Application of double integrals

Exercise

1.

Find the area of the following regions．

(a)

The region enclosed by the curve $${x = \cos^{3}{t}, y = \sin^{3}{t} \ (0 \leq t \leq \frac{\pi}{2})}$$ and $xy$ axes.

(b)

The region enclosed by $${r = a\cos{3\theta} \ (a > 0)}$$.

(c)

The region enclosed by $${y = \frac{8}{x^2 + 4}}$$ and $${y = \frac{x^2}{4}}$$.

2.

Find the surface area of the following region.．

(a)

A sphere $${x^2 + y^2 + z^2 = a^2}$$ with the radius $a$.

(b)

The region $${x^2 + y^2 \leq a^2}$$ corresponds to $z = xy$.

(c)

The region common to the cylinders $${x^2 + z^2 = a^2}$$ and $${x^2 + y^2 = a^2}$$.

(d)

The surface created by evolving $y = mx \ (0 \leq x \leq k)$ with $(m > 0)$ around $x$ axis.

3.

Find the volume of the following solids.

(a)

The cylinder $${x^2 + y^2 \leq a^2}$$ with $0 \leq z \leq x$.

(b)

The closed region $${0 \leq z \leq 1 - x^2, x \leq 1 - y^2, x \geq 0, y \geq 0}$$.

(c)

The sphere $${x^2 + y^2 + z^2 \leq a^2}$$ common to $${x^2 + y^2 \leq ax}$$.

(d)

The region enclosed by the conical surface $${z = 1 - \sqrt{x^2 + y^2}}$$ and the planes $z = x$ and $x= 0$.