The Problem of Areas

Q.

If $\cos \theta =\frac{8}{17}$ and $\theta$ lies in the $1$st quadrant, then the value of $\cos (30°+\theta )+\cos (45°-\theta )+\cos (120°-\theta )$ is

1. $\left(\frac{23}{17}\right)\left[\frac{(\sqrt{3}–1)}{2}+\frac{1}{\sqrt{2}}\right]$

2. $\left(\frac{23}{17}\right)\left[\frac{(\sqrt{3}+1)}{2}+\frac{1}{\sqrt{2}}\right]$

3. $\left(\frac{23}{17}\right)\left[\frac{(\sqrt{3}–1)}{2}-\frac{1}{\sqrt{2}}\right]$

4. $\left(\frac{23}{17}\right)\left[\frac{(\sqrt{3}+1)}{2}-\frac{1}{\sqrt{2}}\right]$

Q. The area bounded by the curve $y=-4x+3$ and $y=0$ between $x=2$ and $x=4$ is

1. $-20$
2. $26$
3. $-18$
4. $18$

Q.

Find the maximum area $(insq.units)$ of a rectangle having its base on the $x-axis$ and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola.

Q. Find the area of the region bounded$x^2=16y,y=1,y=4$ and the y - axis in the first quadrant.

Q. Calculate the area bounded by the curve $y=x$ and the $x$ - axis from $x=1$ to $\sqrt{5}$.

1. $\frac{1}{2}$
2. 2
3. 1
4. $\frac{\sqrt{5}}{2}$

Q.

How can a string of length $l$ be made into a rectangle so as to maximize the area of the rectangle?

1. $\frac{l^2}{16}$

2. $\frac{l^2}{4}$

3. $\frac{l^2}{8}$

4. $\frac{l^2}{2}$

Q. Calculate the area bounded by curve $y=x$ and x-axis from x = 1 to $\sqrt{5}$ .

Q. If $x=(3y^2+4y+3)$, then $\int xdy$ will be

1. $3y^2+4y+3+C$
2. $3y^2+8y+3+C$
3. $y^3+2y^2+3y+C$
4. $3y^3+4y+6+C$

Q.

Find the dimensions of a rectangle with perimeter $68ft$ whose area is as large as possible.

Q. Find the area of the region bounded by the curve $y^2=4x,x=1$, $x=4$ and the $x$-axis in the first quadrant.

1. $4$
2. $\frac{28}{3}$
3. $\frac{7}{3}$
4. $\frac{4}{3}$

Q. ${\left(\tan x+\cot x\right)}^2={\sec }^{2}x+\cos e{c}^{2}x$

Q. Find the area of the region bounded by $x^2=16y,y=1,y=4$ and the $y$-axis in the first quadrant.

1. $\frac{8}{3}$
2. $\frac{56}{3}$
3. $\frac{64}{3}$
4. $\frac{48}{3}$

Q. The area of the region bounded by the curve $y=9-x^2$, $x$-axis and the lines $x=0$ and $x=3$ is ______

1. $9$
2. $27$
3. $18$
4. $36$

Q. Calculate the area bounded by the curve $y=x$ and the $x$ - axis from $x=1$ to $\sqrt{5}$.

1. $1$
2. $\frac{1}{2}$
3. $2$
4. $\frac{\sqrt{5}}{2}$

Q. Find the area of the region bounded by $x^2=16y,y=1,y=4$ and the $y$-axis in the first quadrant.

1. $\frac{8}{3}$
2. $\frac{56}{3}$
3. $\frac{64}{3}$
4. $\frac{48}{3}$

Q.

How can a string of length $l$ be made into a rectangle so as to maximize the area of the rectangle?

1. $\frac{l^2}{16}$

2. $\frac{l^2}{8}$

3. $\frac{l^2}{4}$

4. $\frac{l^2}{2}$

Q. If $\frac{d\tan x}{dx}={\sec }^{2}x$, then $\int \frac{2}{1-{\sin }^{2}x}dx$ is equal to:

1. $2\tan x+c$
2. $2{\cos }^{2}x+c$
3. ${\tan }^{2}x+c$
4. $2{\sec }^{2}x+c$

Q. Find the area of the region bounded by the curve $y^2=4x,x=1$, $x=4$ and the $x$-axis in the first quadrant.

1. $\frac{28}{3}$
2. $4$
3. $\frac{7}{3}$
4. $\frac{4}{3}$

Q. Find the area of the region bounded by $x^2=16y,y=1,y=4$ and the $y$-axis in the first quadrant.

1. $\frac{8}{3}$
2. $\frac{56}{3}$
3. $\frac{64}{3}$
4. $\frac{48}{3}$

Q. Find the area of the region bounded by $y=x^3,y=0,x=2$ and $x=4$.

1. 60
2. 16
3. 64
4. 256

Q. If $x=(3y^2+4y+3)$, then $\int xdy$ will be

1. $3y^2+4y+3+C$
2. $3y^2+8y+3+C$
3. $y^3+2y^2+3y+C$
4. $3y^3+4y+6+C$

Q.

How can a string of length $l$ be made into a rectangle so as to maximize the area of the rectangle?

1. $\frac{l^2}{16}$

2. $\frac{l^2}{8}$

3. $\frac{l^2}{4}$

4. $\frac{l^2}{2}$

Q. The area of the region enclosed by the curves $y=x,x=e,y=\frac{1}{x}$ and the positive x-axis is:-

Q. Evaluate:

1. $\ln |\sec x|+c$
2. $\ln |\cos x|+c$
3. $-\ln |\sin x|+c$
4. $\ln |\sin x|+c$

Q. Find the area of the region bounded by the curve $y^2=4x,x=1$, $x=4$ and the $x$-axis in the first quadrant.

1. $\frac{28}{3}$
2. $4$
3. $\frac{7}{3}$
4. $\frac{4}{3}$

Q. $\int \frac{\ln \left(\tan x\right)}{\sin x\cos x}dx$ is equal to

1. $\frac{1}{2}\left(\ln \right(\tan x){)}^2+c$
2. None of these
3. $\frac{1}{2}\ln \left(\tan x\right)+c$
4. $\frac{1}{2}\ln \left({\tan }^{2}x\right)+c$

Q. If $x=(3y^2+4y+3)$, then $\int xdy$ will be

1. $3y^2+4y+3+C$
2. $3y^2+8y+3+C$
3. $y^3+2y^2+3y+C$
4. $3y^3+4y+6+C$

Q. If $\frac{d\tan x}{dx}={\sec }^{2}x$, then $\int \frac{2}{1-{\sin }^{2}x}dx$ is equal to:

1. $2\tan x+c$
2. $2{\cos }^{2}x+c$
3. ${\tan }^{2}x+c$
4. $2{\sec }^{2}x+c$