Evaluate the following:
\(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx } \)

The region enclosed by the circle x² + y² = a² is divided into two segments by the line x = h. Find the area of the smaller segment.

Find, by integration, the area of the region bounded by the lines 5x − 2y = 15, x + y + 4 = 0 and the x-axis

Father of a family wishes to divide his square field bounded by x = 0, x = 4, y = 4 and y = 0 along the curve y² = 4x and x² = 4y into three equal parts for his wife, daughter and son. Is it possible to divide? If so, find the area to be divided among them.

Find, by integration, the volume of the container which is in the shape of a right circular conical frustum.