Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\) using the Riemann sums corresponding to 5 subintervals of equal width and applying
(i) left-end rule
(ii) right-end rule
(iii) the mid-point rule.

Evaluate: \(\int ^4_{-4}\) |x+3| dx. 

Show that \(\int ^\frac{\pi}{2}_0\) \(\frac {dx}{4+5 sin x}\) = \(\frac {1}{3}\) log_e 2.

Evaluate : \(\int ^\frac{\pi}{4}_{0} \frac{1}{sin x+cos x}\) dx

Evaluate\(\int ^{\pi}_{0} \frac{x}{1+sin x}\) dx

Evaluate: \(\int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} d x\) 

Prove that \(\int ^\frac{\pi}{4}_{0}\) log(1+tan x)dx = \(\frac{\pi}{8}\) log2.

Show that \(\int ^{1}_{0} (tan ^{-1} x + tan ^{-1}(1-x))\) dx = \(\frac {\pi}{2}\) - log_e2 

Evaluate \(\int ^{3}_{2} \frac{\sqrt {x}}{\sqrt {5-x}+\sqrt {x}}\)dx.

Evaluate \(\int ^{\pi}_{-\pi} \frac{cos ^2 x}{1+ a^x}\) dx

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 1 }{ \frac { log(1+x) }{ 1+{ x }^{ 2 } } } dx\)

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ \pi }{ \frac { xsinx }{ 1+sinx } dx } \)

Evaluate the following integrals using properties of integration:
\(\int _{ \frac { \pi }{ 8 } }^{ \frac { 3\pi }{ 8 } }{ \frac { 1 }{ 1+\sqrt { tanx } } dx } \)

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ \pi }{ x\left[ { sin }^{ 2 }(sinx)+{ cos }^{ 2 }(cosx) \right] } dx\)

Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 1+5{ cos }^{ 2 }x } } \)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 4{ sin }^{ 2 }x+5{ cos }^{ 2 }x } } \)

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 by the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\)

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

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 between the parabola x² = y and the curve y = |x|.

Find the area of the region bounded by y = cos x, y = sin x, the lines x = \(\frac{\pi}{4}\) and x = \(\frac{5\pi}{4}\).

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 the area of the region in the first quadrant bounded by the parabola y² = 4x, the line x + y = 3 and y-axis.

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

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

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

Find the area of the region bounded by the curve 2+x−x²+y = 0 , x-axis, x = −3 and x = 3.

Find the area of the region bounded by the line y = 2x + 5 and the parabola y = x² − 2x.

Find the area of the region bounded between the curves y = sin x and y = cos x and the lines x = 0 and x = \(\pi\)

Find the area of the region bounded by y = tan x, y = cot x and the lines x = 0, x = \(\frac{\pi}{2}\), y = 0

Find the area of the region bounded by the parabola y² = x and the line y = x − 2

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.

The curve y = (x − 2)² +1 has a minimum point at P. A point Q on the curve is such that the slope of PQ is 2. Find the area bounded by the curve and the chord PQ.

Find the area of the region common to the circle x² +  y² = 16 and the parabola y² = 6x.

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

Find, by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = 2x², y = 0 and x = 1.

Find, by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = e−^2x y = 0, x = 0 and x = 1

Find, by integration, the volume of the solid generated by revolving about the y-axis, the region enclosed by x² = 1+ y and y = 3.

The region enclosed between the graphs of y = x and y = x² is denoted by R, Find the volume generated when R is rotated through 360° about x-axis.

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

A watermelon has an ellipsoid shape which can be obtained by revolving an ellipse with major-axis 20 cm and minor-axis 10 cm about its major-axis. Find its volume using integration.