Integration: the Area Problem

Key Questions

• Why does integration find the area under a curve?

  Let us look at the definition of a definite integral below.

  Definite Integral
  #\int_a^b f(x) dx=lim_{n to infty}sum_{i=1}^n f(a+iDelta x)Delta x#,
  where #Delta x ={b-a}/n#.

  If #f(x)ge0#, then the definition essentially is the limit of the sum of the areas of approximating rectangles, so, by design, the definite integral represents the area of the region under the graph of #f(x)# above the x-axis.

  Wataru · · Sep 18 2014
• When using integration to find an area, exactly which "area" is found?

  If we consider the general integral:

  # I = int_a^b \ f(x) \ dx #

  Then the Integration calculates the "net" area between the curve #y=f(x)# from #x=a# to #x=b#, and the #x#-axis.

  By "net" area, we consider any area below the curve and above the #x#-axis to be positive , and any area above the curve and below the #x#-axis to be negative.

  Steve M · 1 · May 24 2018

• How do you find the area of a region using integration?
• How do you use integration to find area under curve?
• Why does integration find the area under a curve?
• How do I evaluate #int_0^5|x-5|dx# by interpreting it in terms of areas?
• How do you find the area of the parallelogram with vertices (4,5), (9, 9), (13, 10), and (18, 14)?
• How do you evaluate the integral of absolute value of (x - 5) from 0 to 10 by finding area?
• How do you find the area of the parallelogram with vertices k(1,2,3), l(1,3,6), m(3,8,6), and n(3,7,3)?
• How do you find the area of the parallelogram with vertices: p(0,0,0), q(-5,0,4), r(-5,1,2), s(-10,1,6)?
• How do you evaluate #int5# between the interval [0,4]?
• What is a surface integral?
• What are examples of functions that cannot be integrated?
• How do you determine the values of the coefficients a and b that will give a minimum value of the area, #S#, of the region enclosed by the curve #y=-x^2+ax+b# (which passes through point (1,2)) and the curve #y=(1/2)x^2#?
• How do you find the area bounded by the curve #y= 3-2x-x^2# and the x axis?
• How do you find the region inside cardioid #r=1+cos(theta)# and outside the circle #r=3cos(theta)#?
• How do you find the region inside cardioid #r=(alpha)(1+sin(theta)), (alpha)>0#?
• How do you find the area of the region cut from the second quadrant by the cardioid r=1-cosθ?
• How do you find the area of the region enclosed by the cardioid #r=2+2cos(theta)#?
• How do you find the area inside of the Cardioid #r=3+2cos(theta)# for #0 <= (theta) <= 2pi#?
• How do you find the area inside of the Cardioid #r = 2+2cosθ# and outside the circle r = 3?
• How do you find the area inside of the circle #r = 3sin(theta)# and outside the cardioid #r = 1 + sin(theta)#?
• How do you find region bounded by circle r = 3cosΘ and the cardioid r = 1 + cosΘ?
• How do you find the area bounded by the cardioid #r=1+cos(theta)#?
• How do you find the area of the region shared by the circles #r=2cos(theta)# and #r=2sin(theta)#?
• How do you find the area of the region shared by the cardioid #r=2(1+cos(theta)) #and the circle r=2?
• What is the area under the curve in the interval [-3, 3] for the function #f(x)=x^3-9x^2#?
• What is the area under the curve in the interval [0, 3] for the function #y=x^2+1#?
• What is the area under the curve in the interval [2,7] for the function #1/(1 + x^2)^(1/2) #?
• What is the area under the curve #y = 2x^{-3}# from 6 to 10?
• What is the area under the curve #f(x)=x^2+2# from 1 to 3?
• What is the area under the curve #y=sqrtt# from 0 to 1?
• The area under the curve y=e^-x between x=0 and x=1 is rotated about the x axis find the volume?
• How do you find the area of the region under the given curve y = (2x + 2)^1/2 on the interval [0,1]?
• How do you use integrals to find the area bounded by the curve y = (x^2 - 9) and the x-axis for x = 0 to x = 5?
• How do you find the area #y = f(x) = 4/x^2# , from 1 to 2 using ten approximating rectangles of equal widths and right endpoints?
• Question #8e5b5
• How do you find the integral of #int dx/(sqrt(2x-1)# from 1/2 to 2?
• How do you find the area between the given curve #y= x^2# and the x-axis given in the interval [0,1]?
• How do you find the area between the given curve #y= x^(1/2)# and the x-axis given in the interval [2,5]?
• How do you find the area enclosed by the x-axis and the given curve #y=(6/x)# for x between -4 & -2?
• What is the area under #f(x)=x^3-1# in #x in[0,1] #?
• What is the area under #f(x)=x^2-3x+5# in #x in[0,1] #?
• What is the area under #f(x)=5# in #x in[0,2] #?
• What is the area under #f(x)=5x-1# in #x in[0,2] #?
• What is the net area between #f(x)=sinx# in #x in[0,2pi] # and the x-axis?
• What is the net area between #f(x)=tanx# in #x in[0,pi/3] # and the x-axis?
• What is the net area between #f(x)=ln(2/x)# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=ln(x+1)# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=x/ln(x^2)# in #x in[3,10] # and the x-axis?
• What is the net area between #f(x)=(x-x^2)/ln(x^2)# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=(x-x^2)/ln(x^2+1)# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=(x-3)e^x# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=7x-8# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=2/x-x# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=ln(x^3-x+2)/x^2# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=(x^2-5)/(x^3-5x# in #x in[1,2] # and the x-axis?
• What is the net area between #f(x)=xsinx# in #x in[0,4pi] # and the x-axis?
• What is the net area between #f(x)=sinxcosx# in #x in[0,2pi] # and the x-axis?
• What is the net area between #f(x)=x^3-x^2+5# in #x in[2,5] # and the x-axis?
• What is the net area between #f(x)=x^3+4x^2-2# in #x in[2,5] # and the x-axis?
• What is the net area between #f(x)=-4x^3+2x^2-5# in #x in[1,5] # and the x-axis?
• What is the net area between #f(x)=-x^3-2x^2-x# in #x in[0,3] # and the x-axis?
• There is a line through the origin that divides the region bounded by the parabola ?
• What is the net area between #f(x) = 3x^2-4x+2# and the x-axis over #x in [1, 3 ]#?
• What is the net area between #f(x) = 3x^2-x+2# and the x-axis over #x in [1, 3 ]#?
• What is the net area between #f(x) = 3x^2-x+2# and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = x^2-x+2# and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = e^(3x)-4x# and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = e^(3x)-2x+1# and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = e^(3-x)-2x+1# and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = e^(3-x)-2x+1# and the x-axis over #x in [0, 3 ]#?
• What is the net area between #f(x) = e^(3-2x)-2x+1# and the x-axis over #x in [0, 3 ]#?
• What is the net area between #f(x) = sinx# and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = sinx - cosx# and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = sinx - cos^2x# and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cosx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cosxsinx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cosxsin^2x # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cos^2xsin^2x # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cos^2xsinx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cos2x-sinx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cos2x-xsinx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = xsinx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = x-cosx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = x-sinx # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = x-sin^2x # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = x-4cos^2x # and the x-axis over #x in [0, 3pi ]#?
• What is the net area between #f(x) = cscx -xsinx# and the x-axis over #x in [pi/6, (5pi)/8 ]#?
• What is the net area between #f(x) = cscx -cosxsinx# and the x-axis over #x in [pi/6, (5pi)/8 ]#?
• What is the net area between #f(x) = cotxcscx -sinx# and the x-axis over #x in [pi/6, (5pi)/8 ]#?
• What is the net area between #f(x) = x-xsqrt(4x-3) # and the x-axis over #x in [1, 4 ]#?
• What is the net area between #f(x) = x-xsqrt(4x+1) # and the x-axis over #x in [1, 4 ]#?
• What is the net area between #f(x) = x-sqrt(x+1) # and the x-axis over #x in [1, 4 ]#?
• What is the net area between #f(x) = x/sqrt(x+1) # and the x-axis over #x in [1, 4 ]#?
• What is the net area between #f(x) = 1/sqrt(x+1) # and the x-axis over #x in [1, 4 ]#?
• What is the net area between #f(x) = -sqrt(x+1) # and the x-axis over #x in [1, 4 ]#?
• What is the net area between #f(x) = x^2-sqrt(x+1) # and the x-axis over #x in [1, 7 ]#?
• What is the net area between #f(x) = (x-2)^3 # and the x-axis over #x in [1, 5 ]#?
• What is the net area between #f(x) = xe^x-3x # and the x-axis over #x in [1, 5 ]#?
• What is the net area between #f(x) = 4/x # and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = 2/x^3 # and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = 2/(x+3) # and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = 2/(x+1)^2 # and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = x^2-ln(x^2+1) # and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = -ln(x^2+1) # and the x-axis over #x in [1, 2 ]#?
• What is the net area between #f(x) = -xln(x^2-1) # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = x^2-xln(x^2-1) # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = sqrt(x^2+2x+1) # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = 1/sqrt(x^2+2x+1) # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = sqrtx-x # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = sqrt(x+3)-x^3 # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = sqrt(x+3)-x^3+6x # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = x^3+6/x # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = x^2+1/x # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = x^2+1/x # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = xlnx-xe^x # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = e^(2x)-xe^x # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = x^2-x+8 # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = x+8 # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = x-4 # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = 2x-6 # and the x-axis over #x in [2, 4 ]#?
• What is the net area between #f(x) = 2x-5 # and the x-axis over #x in [2, 3 ]#?
• What is the net area between #f(x) = 3-xsqrt(x^2-1) # and the x-axis over #x in [2, 3 ]#?
• What is the net area between #f(x) = xsqrt(x^2-1) + 6# and the x-axis over #x in [2, 3 ]#?
• Let R be the region in the first quadrant bounded above by the graph of #y=(6x+4)^(1/2)# the line #y=2x# and the y axis, how do you find the area of region R?
• A rectangle has one corner in quadrant 1 on the graph of #y=16-x^2#, another at the origin, and the third on the positive y-axis, and the fourth on the positive x-axis. How do you express the area, A, of the rectangle as a function of x?
• What is the area of the region in the first quadrant by the graph of #y=e^(x/2)# and the line x=2?
• What is the area of the region in the first quadrant enclosed by the graphs of y = cosx, y= x, and the y -axis?
• What's the area of the first quadrant region bounded by the y-axis, the line #y=4-x # and the graph of #y=x-cosx#?
• Let R be the region in the first quadrant bounded by the graphs of #y=x^2#, #y=0#, and #x=2#, how do you find the area of R?
• Let R be the region in the first quadrant enclosed by the graphs of #y=e^(-x^2)#, #y=1-cosx#, and the y axis, how do you find the area of the region R?
• Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=sin(x)# and #y=cos(x)#, how do you find the area of R?
• How do you find the area in the first quadrant between the graphs of #y^2=(x^3)/3# and #y^2=3x#?
• Let R be the region in the first quadrant enclosed by the graph of #y=2cosx#, the x-axis, and the y-axis, how do you find the area of the region R?
• Let R be the first quadrant region enclosed by the graph of #y= 2e^-x# and the line x=k, how od you find the area of R in terms of k?
• Let R be the region in the first quadrant that is enclosed by the graph of #y=tan x#, the x-axis, and the line x=pi/3, how do you find the area?
• What is the area of the region in the first quadrant that is enclosed by the graphs of #y=x^3+8# and #y=x+8#?
• Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=4-x^2# and #y=1+2sinx#, how do you find the area?
• Let f and g be the functions given by #f(x)=e^x# and #g(x)=1/x#. What is the area of the region enclosed by the graphs of f and g between x=1 and x=2?
• How do you find the area of the region between the graphs of #y=x^2# and #y=-x# from x=0 to x=3?
• Let R be the region in the first quadrant bounded by the x-axis, the graph of #x=y^2+2#, and the line x=4. What is a the interval for the area of R?
• Let R be the region in the first and second quadrants bounded above by the graph of #y=20/(1+x^2)# and below by the horizontal line y=2, how do you find the area?
• Let R be the region in the first quadrant bounded by the graphs of #(x^2/ 9) + (y^2 /81)=1# and #3x+y=9#, how do you find the area?
• How do you find the area bounded by the curves #y = -4sin(x)# and #y = sin(2x)# over the closed interval from 0 to pi?
• How do you find the area under the graph of #f(x)=e^(-2lnx)# on the interval [1, 2]?
• How do you use the trapezoidal rule with n = 4 to approximate the area bounded by the curves #y = sin2x#, the lines x = 0, y = 0, and x = 1?
• How do you write the definite integral to find the smaller area cut from the circle #x^2 + y^2 = 25# by the line x = 3?
• How do you find the volume when the region bounded by y = x+3, y = 0, x = -3 and x = 3 is revolved around the x-axis?
• How do you find the area enclosed by the curve #y^2 = x(1 - x)#?
• Let f and g be the functions given by #f(x)=1+sin(2x)# and #g(x)=e^(x/2)#. Let R be the region in the first quadrant enclosed by the graphs of f and g. How do you find the area?
• How do you find the exact value of the area in the first quadrant enclosed by graph of y=sinx and y=cosx?
• How do you find the area of the region bounded by the given curves #y = 6x^2lnx # and #y = 24lnx#?
• Question #cb563
• Question #fce84
• How do you find the area enclosed by graphs of #y=sqrt [x]# and #y= x^3#?
• Let r be the region bounded by the graphs of #y= sqrt(x)# and #y=x/2#, how do you find the area of R?
• How do we find the area inside the cardioid #f(theta)=a(1+costheta)#?
• Question #c62e7
• Find b, c and d so that the quadrilateral is a parallelogram with area equal to 80 square units?
• Let R be a region in he first quadrant bounded above by #y=x+2# and below by #y=e^x#, how do you find the area of R?
• What is the area of the largest rectangle that can be inscribed under the graph of y=2 cos x for -π /2 ≤x ≤π /2?
• Question #ad67f
• Question #d9bfd
• Question #75f31
• What is the area between the line #y = x# and the curve #y = x^3#?
• Question #40f71
• How do you use the limit process to find the area of the region between the graph #y=-2x+3# and the x-axis over the interval [0,1]?
• How do you use the limit process to find the area of the region between the graph #y=x^2+2# and the x-axis over the interval [0,1]?
• How do you use the limit process to find the area of the region between the graph #y=x^2+1# and the x-axis over the interval [0,3]?
• How do you use the limit process to find the area of the region between the graph #y=16-x^2# and the x-axis over the interval [1,3]?
• How do you use the limit process to find the area of the region between the graph #y=1-x^3# and the x-axis over the interval [-1,1]?
• How do you use the limit process to find the area of the region between the graph #y=64-x^3# and the x-axis over the interval [1,4]?
• How do you use the limit process to find the area of the region between the graph #y=2x-x^3# and the x-axis over the interval [0,1]?
• How do you use the limit process to find the area of the region between the graph #y=x^2-x^3# and the x-axis over the interval [-1,0]?
• How do you use limits to find the area between the curve #y=x^2# and the x axis from [0,5]?
• How do you use limits to find the area between the curve #y=2x^3# and the x axis from [1,5]?
• How do you use limits to find the area between the curve #y=x^4# and the x axis from [0,5]?
• How do you use limits to find the area between the curve #y=x^2+6x# and the x axis from [0,4]?
• How do you use limits to find the area between the curve #y=x^2-x+1# and the x axis from [0,3]?
• How do you use limits to evaluate #int 8xdx# from [0,2]?
• How do you use limits to evaluate #int (x+2)dx# from [1,4]?
• How do you use limits to evaluate #int x^2dx# from [0,4]?
• How do you use limits to evaluate #int8x^3dx# from [3,5]?
• How do you use limits to evaluate #int(x^2+4x-2)dx# from [1,4]?
• How do you use limits to evaluate #int(x^5+x^2)dx# from [0,2]?
• What is the area below the curve #f(x)=x^3 -4x^2 +2x+ 8# and bounded by the y-axis, the x axis and x=3 expressed as an integral?
• Question #e8898
• How do you find the area of region bounded by the graphs of y +x= 6 and y +2x-3=0?
• How do you find the area of the region under the curve #y=4x^-2# from x=1, to #x=oo#?
• How do you find the area of the region under the curve #y=1/sqrt(2x-1)# from x=1/2, to x=1?
• Let #R# be the region in the first quadrant bounded by the #x# and #y# axis and the graphs of #f(x) = 9/25 x +b# and #y = f^-1 (x)#. If the area of #R# is 49, then the value of #b#, is ? A) #18/5# B) #22/5# C) #28/5# D) none
• Given #f(x)# is a polynomial function of #x#, satisfying #f(x)*f(y)= f(x) +f(y)+f(x y)-2# and that #f(2)=5#. Then #f(3)# is equal to? A) 10 B) 24 C) 15 D) none
• What is the area bounded by the curves? : # 4x + y^2 = 32 # and # x=y #
• How do you use integrals?
• Find the area bounded by # x = -y^2 # and # y = x+2# using a double integral?
• Find the surface area of a sphere of radius r?
• Alex and Veronica were discussing the definite integral #int_0^3 (x^2 − 1)dx#. Alex said it represented the total area bounded by #f(x) = x^2 - 1# and the #x#-axis , between #x = 0# and #x = 3#. Veronica said the total area was larger?
• How do i answer this? # int_2^4 \ (2x)/(x^2+1) # via a Riemann sum.
• Write the Riemann sum to find the area under the graph of the function #f(x) = x^2# from #x = 1# to #x = 5#?
• A firm manufacturing 2000 items.It estimated that the rate of change of production P with the respect to additional number of workers x is given by dP/dx=100-12x^.5.If the firm employees 25 more workers,then the new level of production of the items is?