Property 7

Q. The value of ${\int }_{-\pi }^{\pi }{\sin }^{3}x{\cos }^{2}x\mathrm{dx}$ is equal to

Q.

Area bounded by curve $y=x^3,x-axis$ axis and ordinates x=1 and x=4, is

1. $64sq.unit$

2. $27sq.unit$

3. $\frac{127}{4}sq.unit$

4. $\frac{255}{4}sq.unit$

Q.

Area between the curve y = cos x and x- axis when 0 $\le$ x is [MP PET 1997]

1. 2

2. 4

3. 3
4. \N

Q.

Area bounded by parabola $y^2=x$ and straight line 2y = x is

1. $\frac{4}{3}$
2. $1$

3. $\frac{2}{3}$

4. $\frac{1}{3}$

Q.

Area bounded by curve y = k sin x between x = $\pi$ and x = $x=2\pi$ is

1. 2k sq. unit
2. \N

3. $\frac{k^2}{2}$ sq. unit

4. k sq. unit

Q.

Area under the curve y = sin 2x + cos 2x between x=0 and $x=\frac{\pi }{4}$is

1. 2 sq. unit

2. 1 sq. unit

3. 3 sq. unit

4. 4 sq. unit

Q.

Area bounded by curve xy = c, x - axis between x=1 and x=4, is

1. c log 3 sq. unit
2. 2 log c sq.unit

3. 2 c log 2 sq. unit

4. 2c log 5 sq. unit

Q.

Area bounded by the curve y = sin x between x = 0 and x = 2$\pi$ is

1. 2 sq. unit

2. 4 sq. unit

3. 8 sq. unit

4. None of these

Q. Let $f:R\to Randg:R\to R$ be continuous functions. Then, the value of the integral
${\int }_{-\pi }^{\pi }\left(f\left(x\right)+f(-x)\right]\left(g\left(x\right)-g(-x)\right]\mathrm{dx}$ is

1. \N
2. 1
3. $\frac{\pi }{2}$
4. $\frac{\pi }{4}$

Q. Find the area enclosed between sin(x) & cos(x) between x = 0 & $x=\frac{\pi }{2}$.

1. $2\sqrt{2}$
2. $2$
3. $2\sqrt{2}-2$
4. $2\sqrt{2}+2$

Q. $\begin{array}{cc}\text{List I}& \text{List II}\\ \text{(I)}{\int }_{-1}^1\frac{dx}{(1+x^2)(1+3^x)}\text{is}& \text{(P)}\frac{\pi }{2}\\ \text{(II)}7\pi {\int }_0^1(1-x^2{)}^{\frac{3}{2}}{x}^{3}dx\text{is}& \text{(Q)}\frac{\pi }{4}\\ \text{(III)}\frac{\pi }{4}.\frac{{\int }_0^{10}\left[x\right]dx}{{\int }_0^{10}\left\{x\right\}dx},\text{(where [.] {.} denotes}& \text{(R)}\frac{9\pi }{4}\\ \text{the greatest integer and}& \\ \text{the fractional part function)}& \\ \text{(IV)}{\int }_0^{\frac{\pi }{4}}[\tan x+[\sin x+[\cos x+[\sec x\left]\right]\left]\right]dx,& \text{(S)}\frac{2\pi }{5}\\ \text{(where [.] denotes the greatest integer function)}& \end{array}$

Which of the following is only CORRECT combination?

1. (I)-(S)
2. (II)-(R)
3. (III)-(R)
4. (IV)-(P)

Q. If the area function ${\int }_a^{x}f\left(x\right)dx=x^2-a^2$ then f(x)is -

1. $\frac{x^3}{3}$
2. 2x
3. $3x^3$
4. none of these

Q.

Area under the curve $y=\sqrt{3x+4}$between x=0 and x=4, is

1. $\frac{56}{9}sq.unit$

2. $\frac{64}{9}sq.unit$

3. 8 sq. unit

4. None of these

Q.

Area bounded by y = x sin x and x - axis between x=0 and x = 2$\pi$ is [Roorkee 1981; RPET 1995]

1. \N
2. 2 $\pi$ sq. unit
3. $\pi$ sq. unit
4. 4 $\pi$ sq. unit

Q.
If area of shaded portion in figure A is $A_1$ and that of shaded portion in figure B is $A_2$. Then,

1. $A_1>A_2$
2. $A_1<A_2$
3. $A_1=A_2$
4. $Insufficientinformation$

Q. If the value of the definite integral
$\underset{-1}{\overset{1}{\int }}{\cot }^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)\cdot {\cot }^{-1}\left(\frac{x}{\sqrt{1-(x^2{)}^{\left|x\right|}}}\right)dx=\frac{{\pi }^2(\sqrt{a}-\sqrt{b})}{\sqrt{c}},$ where $a,b,c\in N$ in their lowest form, then the value of $(a+b+c)$ is

Q. The value of ${\int }_{-\pi }^{\pi }{\sin }^{3}x{\cos }^{2}x\mathrm{dx}$ is equal to

Q. If two functions f(x) and g(x) defined on R intersect each other at x = a & x =b then the area enclosed between these two functions will be -

1. ${\int }_a^b\left[f\right(x)-g(x\left)\right]dx$
2. ${\int }_a^b\left[g\right(x)-f(x\left)\right]dx$
3. ${\int }_a^b\left[f\right(x)-g(x\left)\right]|dx$
4. $Noneofthese$