The Problem of Areas

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. 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 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 $y=x^3,y=0,x=2$ and $x=4$.

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

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

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

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