Chapter 13

Find the area of the region(s) enclosed by the curve \(f(x)=x^{3},\) the \(x\) axis, and the lines \(x=-1\) and \(x=2\).

Find the area of the region(s) enclosed by the curve \(y=12 x-61,\) the \(x\) -axis, and the lines \(x=0\) and \(x=4\).

Find the approximate area under the curve \(f(x)=\frac{1}{x}\) from \(x=1\) to \(x\) \(=5,\) using four right-endpoint rectangles of equal lengths.

Find the approximate area under the curve \(y=x^{2}+1\) from \(x=0\) to \(x=\) \(3,\) using the Trapezoidal Rule with \(n=3\).

Find the area of the region bounded by the graphs of all four equations: \(f(x)=\sin \left(\frac{x}{2}\right) ; x\) -axis; and the lines, \(x=\frac{\pi}{2}\) and \(x=\pi .\)

Find the volume of the solid obtained by revolving about the \(x\) -axis, the region bounded by the graphs of \(y=x^{2}+4,\) the \(x\) -axis, the \(y\) -axis, and the lines \(x=3\).

The area under the curve \(y=\frac{1}{x}\) from \(x=1\) to \(x=k\) is 1 . Find the value of \(k\).

Find the volume of the solid obtained by revolving about the \(y\) -axis the region bounded by \(x=y^{2}+1, x=0, y=-1,\) and \(y=1\).

Find the volume of the solid obtained by revolving about the \(x\) -axis the region bounded by the graphs of \(f(x)=x^{3}\) and \(g(x)=x^{2}\).

The base of a solid is a region bounded by the circle \(x^{2}+y^{2}=4\). The cross sections of the solid perpendicular to the \(x\) -axis are equilateral triangles. Find the volume of the solid.

Find the volume of the solid obtained by revolving about the \(y\) -axis, the region bounded by the curves \(x=y^{2}\) and \(y=x-2\).

The function \(f(x)\) is continuous on \([0,12],\) and the selected values of \(f(x)\) are shown in the table. $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline f(x) & 1 & 2.24 & 3 & 3.61 & 4.12 & 4.58 & 5 \\ \hline \end{array} $$ Find the approximate area under the curve of \(f\) from 0 to 12 using three midpoint rectangles.

$$ \int_{-a}^{a} e^{x^{1}} d x=k, \text { find } \int_{0}^{a} e^{x^{2}} d x \text { in terms of } k $$

(Calculator) Find a point on the parabola \(y=\frac{1}{2} x^{2}\) that is closest to the point (4,1)

The velocity function of a particle moving along the \(x\) -axis is \(v(t)=t\) \(\cos \left(t^{2}+1\right)\) for \(t \geq 0\) (a) If at \(t=0,\) the particle is at the origin, find the position of the particle at \(t=2\). (b) Is the particle moving to the right or left at \(t=2 ?\) (c) Find the acceleration of the particle at \(t=2\) and determine if the velocity of the particle is increasing or decreasing. Explain why.