Let $$f\left( x \right)$$ be a given integrable function such that $$f\left( {x + k} \right) = f\left( x \right)$$ for all $$x\, \in \,R.$$ Then $$\int_a^{a + k} {f\left( x \right)dx} $$ depends for its value on :
A.
$$a$$ only
B.
$$k$$ only
C.
both $$a$$ and $$k$$
D.
neither $$a$$ nor $$k$$
Answer :
$$k$$ only
Solution :
$$\eqalign{
& \int_a^{a + k} {f\left( x \right)} dx = \int_0^{a + k} {f\left( x \right)} dx - \int_0^a {f\left( x \right)} dx \cr
& = \int_0^k {f\left( x \right)} dx + \int_k^{a + k} {f\left( x \right)} dx - \int_0^a {f\left( x \right)} dx \cr
& {\text{For the second integral, put }}x = z + k \cr
& {\text{Then }}\int_k^{a + k} {f\left( x \right)} dx = \int_0^a {f\left( {z + k} \right)} dz = \int_0^a {f\left( z \right)} dz \cr
& \therefore \int_a^{a + k} {f\left( x \right)} dx = \int_0^k {f\left( x \right)} dx + \int_0^a {f\left( x \right)} dx - \int_0^a {f\left( x \right)} dx \cr
& = \int_0^k {f\left( x \right)} dx \cr
& {\text{which depends on }}k{\text{ but not on }}a. \cr} $$
Releted MCQ Question on Calculus >> Application of Integration
Releted Question 1
The area bounded by the curves $$y = f\left( x \right),$$ the $$x$$-axis and the ordinates $$x = 1$$ and $$x = b$$ is $$\left( {b - 1} \right)\sin \left( {3b + 4} \right).$$ Then $$f\left( x \right)$$ is-