Evaluate-the-Integral-pi-2-pi-2-0-3-cos-r-2-sin-2-dr-d-

Question Number 37912 by gunawan last updated on 19/Jun/18

Evaluate : the Integral

\int_{- \frac{π}{2}}^{\frac{π}{2}} \int_{0}^{3 \cos θ} r^{2} \sin^{2} θ . d r d θ

Commented by math khazana by abdo last updated on 19/Jun/18

I = \int_{- \frac{π}{2}}^{\frac{π}{2}} A (θ) {s i n}^{2} θ d θ w i t h

A (θ) = \int_{0}^{3 c o s θ} r^{2} d r = \frac{1}{3} {[r^{3}]}_{0}^{3 c o s θ} = \frac{1}{3} 27 {c o s}^{3} θ

= 9 {c o s}^{3} θ \Rightarrow I = 9 \int_{- \frac{π}{2}}^{\frac{π}{2}} {s i n}^{2} θ {c o s}^{3} θ d θ

=_{s i n θ = t} \int_{- 1}^{1} t^{2} (1 - t^{2}) \sqrt{1 - t^{2}} \frac{d t}{\sqrt{1 - t^{2}}}

= \int_{- 1}^{1} (t^{2} - t^{4}) d t = 2 {[\frac{t^{3}}{3} - \frac{t^{5}}{5}]}_{0}^{1}

= 2 {\frac{1}{3} - \frac{1}{5}} = 2 \frac{2}{15} = \frac{4}{15} \Rightarrow I = 9 . \frac{4}{15} = \frac{12}{5}

I = \frac{12}{5} .

Answered by tanmay.chaudhury50@gmail.com last updated on 19/Jun/18

\int_{\frac{- Π}{2}}^{\frac{Π}{2}} {s i n}^{2} θ \times ∣ \frac{r^{3}}{3} ∣_{0}^{3 c o s θ} d θ

\frac{1}{3} \int_{\frac{- Π}{2}}^{\frac{Π}{2}} {s i n}^{2} θ \times 27 {c o s}^{3} θ d θ

9 \int_{- \frac{Π}{2}}^{\frac{Π}{2}} {s i n}^{2} θ . (1 - {s i n}^{2} θ) c o s θ d θ

t = s i n θ d t = c o s θ d θ

9 \int_{- 1}^{1} (t^{2} - t^{4}) d t

9 ∣ \frac{t^{3}}{3} - \frac{t^{5}}{5} ∣_{- 1}^{1}

9 {(\frac{1}{3} - \frac{1}{5}) - (\frac{- 1}{3} - \frac{- 1}{5})}

9 {\frac{1}{3} - \frac{1}{5} + \frac{1}{3} - \frac{1}{5}}

18 (\frac{1}{3} - \frac{1}{5}) = 18 \times \frac{2}{15} = \frac{12}{5}