Given int_0^3f(x)dx=6 and int_0^2f(x)dx=5. Find the value k when int_3^2(f(x)-k)dx=5k. How to do this question? Help needed

2 Answers
Sean
Feb 4, 2018

k=-1/4

Explanation:

.

int_0^3f(x)dx=(F(x))_0^3=F(3)-F(0)=6

int_0^2f(x)dx=(F(x))_0^2=F(2)-F(0)=5

If we subtract the first equation from the second one we get:

F(2)-F(0)-F(3)+F(0)=5-6=-1

F(2)-F(3)=-1

int_3^2(f(x)-k)dx=(F(x)-kx)_3^2=F(2)-2k-(F(3)-3k)=5k

F(2)-F(3)-2k+3k=5k

-1+k=5k

-1=4k

k=-1/4

Ratnaker Mehta
Feb 4, 2018

-1/4.

Explanation:

Prerequisites :

(1):int_a^bphi(x)dx=int_a^cphi(x)dx+int_c^bphi(x)dx; altcltb.

(2):int_a^bphi(x)dx=-int_b^aphi(x)dx; altb.

Given that, int_0^3f(x)dx=6,

rArr int_0^2f(x)dx+int_2^3f(x)dx=6.........................[because, (1)].

rArr 5+int_2^3f(x)dx=6..................................[because," Given]".

rArr int_2^3f(x)dx=1," or, by (2), "int_3^2f(x)dx=-1.......(ast).

Now, int_3^2(f(x)-k)dx=5k,

rArrint_3^2f(x)dx-int_3^2kdx=5k,

rArr -1-kint_3^2dx=5k...[because, (ast)],

rArr -1-k[x]_3^2=5k,

rArr -1-k[2-3]=5k,

rArr -1-k(-1)=5k,

rArr -1=5k-k=4k.

:. k=-1/4.

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