If f(x) is a continuous periodic function with period T, then the… - Vidyadip

MCQ

If $f(x)$ is a continuous periodic function with period $T,$ then the integral $I = \int_a^{a + T} {f(x)\,dx} $ is

A
Equal to $2a$
B
Equal to $3a$
✓
Independent of $a$
D
None of these

✓

Answer

Correct option: C.

Independent of $a$

c
(c) Consider the function $g(a) = \int_a^{a + T} {f(x)dx} $

$ = \int_a^0 {f(x)dx + \int_0^T {f(x)dx + \int_T^{a + T} {\,\,f(x)dx} } } $

Putting $x - T = y$ in last integral, we get

$\int_T^{a + T} {f(x)dx = \int_0^a {f(y + T)dy = \int_0^a {f(y)dy} } } $

==> $g(a) = \int_a^0 {f(x)dx + \int_0^1 {f(x)dx + \int_0^a {f(x)dx} } } $$ = \int_0^T {f(x)dx} $

Hence $ g(a)$ is independent of $a.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $f(x) = \left\{ {\begin{array}{*{20}{c}}
{x,\,\,\,\,\,\,x \in Q}\\
{0,\,\,\,\,\,\,x \notin Q}
\end{array}} \right.;g(x) = \left\{ {\begin{array}{*{20}{c}}
{x,\,\,\,\,\,\,x \in Q}\\
{0,\,\,\,\,\,\,x \notin Q}
\end{array}} \right.$ then function $(f -g)$ is :-

The value of $k$ so that the function $f(x) = \left\{ \begin{array}{l}k(2x - {x^2}),\;\;\;{\rm{when\,}}\,x < 0\\\,\,\,\,\,\,\,\,\,\cos x,\,\,\,\,\,\,{\rm{when\,}}\,x \ge {\rm{0}}\end{array} \right.$ is continuous at $x = 0$, is

Which of the following term is used in a linear programming problem?

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$ Then $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$ is equal to

If $\tan (x + y) + \tan (x - y) = 1,$ then ${{dy} \over {dx}} = $

Let $f(x)=\int_{0}^{x} e^{t} f(t) d t+e^{x}$ be a differentiable function for all $x \in R$. Then $f(x)$ equals ..... .

If $\text{f(x)}=\text{x}\sin\frac{1}{\text{x}},\text{ x}\neq0,$ then the value of the function at x = 0, so that the function is continuous at x = 0, is:

0
-1
1
indeterminate

The area of the region enclosed by the parabola $y=4 x-x^2$ and $3 y=(x-4)^2$ is equal to

The value of f(0), so that the function $\text{f(x)}=\frac{\sqrt{\text{a}^2+\text{ax+x}^2}-\sqrt{\text{a}^2+\text{ax+x}^2}{}}{\sqrt{\text{a+x}}-\sqrt{\text{a-x}}}$ becomes continuous for all x, given by:

$\text{a}^{\frac{3}{2}}$
$\text{a}^{\frac{1}{2}}$
$-\text{a}^{\frac{1}{2}}$
$-\text{a}^{\frac{3}{2}}$

The area bounded by the curve y = 4x - x² and the x-axis is:

$\frac{30}{7}\text{ sq. units}$
$\frac{31}{7}\text{ sq. units}$
$\frac{32}{3}\text{ sq. units}$
$\frac{34}{3}\text{ sq. units}$