Integral Calculus

Locate the centroid of the plane area bounded by the equation \(y^2=4x\), \(x=1\) and the x-axis on the first quadrant.

Find the length of arc of the parabola \(x^2=4y\) from \(x=-2\) to \(x=2\).

Find the surface area (in square units) generated by rotating the parabola arc \(y=x^2\) about the x-axis from \(x=0\) to \(x=1\).

The area enclosed by the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) is revolved about the line \(x=3\). What is the volume generated?

The area in the second quadrant of the circle \(x^2+y^2=36\) is revolved about the line \(y+10=0\). What is the volume generated?

The area bounded by the curve \(y^2=12x\) and the line \(x=3\) is revolved about the line \(x=3\). What is the volume generated?

Given the area in the first quadrant bounded by \(x^2=8y\), the line \(y-2=0\) and the y-axis. What is the volume generated when the area is revolved about the line \(y-2=0\)?

Find the volume (in cubic units) generated by rotating a circle \(x^2+y^2+6x+4y+12=0\) about the y-axis.

Given the area in the first quadrant by \(x^2=8y\), the line \(x=4\) and the x-axis. What is the volume generated by revolving this area about the y-axis.

Find the moment of inertia, with respect to x-axis of the area bounded by the parabola \(y^2=4x\) and the line \(x=1\).

Evaluate \(\displaystyle\iint 2r^2\sin\theta\cos^2\theta\,dr\,d\theta\) for \(0 \le r \le \sin\theta\), \(0 \le \theta \le \dfrac{\pi}{2}\).

Find the volume of the solid formed by revolving the area bounded by the curve \(y^2 = x^3(1 - x)\) in the first quadrant about the x-axis.

A particle moves on a straight line with a velocity \(v = (4 - 2t)^3\) at time t. Find the distance traveled from t = 0 to t = 3.

The area enclosed by the ellipse \(4x^2 + 9y^2 = 36\) is revolved about the line x = 3. What is the volume generated?

Find the area of one loop of \(r^2 = 16\sin 2\theta\).

Find the centroid of the upper half of the circle \(x^2 + y^2 = 9\).

Find the length of the arc of \(6xy = x^4 + 3\) from x = 1 to x = 2.

Find the area bounded by \(y = 6x - x^2\) and \(y = x^2 - 4x\).

If \(f'(x) = \sin x\) and \(f(\pi) = 3\), then \(f(x) =\)

The vertical end of a water trough is an isosceles triangle with a width of 6 feet and a depth of 3 feet. Find the force on one end when the trough is filled with water.