Question # 01:
Evaluate the double integral
Question # 02:
Prove that
Question # 03:
Use double integral to find the volume under the surface given by
over the region bounded by the curves
plz help me.
Question # 01:
Evaluate the double integral
Question # 02:
Prove that
Question # 03:
Use double integral to find the volume under the surface given by
over the region bounded by the curves
plz help me.
1. $\displaystyle \int_{-1}^1{\int_{x^2}^1{(x^2 + y^2)\,dy}\,dx}$
$\displaystyle = \int_{-1}^1{\left[x^2y + \frac{1}{3}y^3\right]_{y=x^2}^{y =1}\,dx}$
$\displaystyle = \int_{-1}^1{\left[x^2 + \frac{1}{3}\right] - \left[x^4 + \frac{1}{3}x^6\right]\,dx}$
$\displaystyle = \int_{-1}^1{\frac{1}{3} + x^2 - x^4 - \frac{1}{3}x^6\,dx}$
$\displaystyle = \left[\frac{1}{3}x + \frac{1}{3}x^3 - \frac{1}{5}x^5 - \frac{1}{21}x^7\right]_{-1}^1$
$\displaystyle = \left[\frac{1}{3} + \frac{1}{3} - \frac{1}{5} - \frac{1}{21}\right] - \left[-\frac{1}{3} - \frac{1}{3} + \frac{1}{5} + \frac{1}{21}\right]$
$\displaystyle = \frac{4}{3} - \frac{2}{5} - \frac{2}{21}$.
I'll let you do the simplifying.