Prove that \(\int _{ 0 }^{ \infty }{ { e }^{ -x }{ x }^{ n }dx=n! } \) where n is a positive integer.

Find the area of the region bounded by the line 6x + 5y = 30, x − axis and the lines x = −1 and x = 3.

Find the area of the region bounded by the line 7x − 5y = 35, x−axis and the lines x = −2 and x = 3.

Find the area of the region bounded between the parabola y² = 4ax and its latus rectum.

Find the area of the region bounded by the y-axis and the parabola x = 5 − 4y − y².

Find the area of the region bounded by x−axis, the sine curve y = sin x, the lines x = 0 and x = 2\(\pi\).

Find the area of the region bounded by 3x − 2y + 6 = 0 , x = −3, x = 1 and x-axis.

Find the volume of a sphere of radius a.

Find the volume of the spherical cap of height h cut of from a sphere of radius r.

Find the volume of the solid formed by revolving the region bounded by the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\), a>b about the major axis.