Problem 69
Evaluate the following integrals. \(\int_{-1}^{1}\left(x^{3}-2 x^{2}+4 x\right) d x\)
Problem 70
Evaluate the following integrals. \(\int_{0}^{4} \frac{3 t}{\sqrt{1+6 t^{2}}} d t\)
Problem 71
The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. Compute the right endpoint estimates \(R_{50}\) and \(R_{100}\) of \(\int_{-3}^{5} \frac{1}{2 \sqrt{2 \pi}} e^{-(x-1)^{2} / 8}\).
Problem 71
Evaluate the following integrals. \(\int_{\pi / 3}^{\pi / 2} 2 \sec (2 \theta) \tan (2 \theta) d \theta\)
Problem 73
Find the antiderivative. \(\int \frac{d x}{(x+4)^{3}}\)
Problem 73
If the half-life of seaborgium- 266 is \(360 \mathrm{~ms}\), then \(k=(\ln (2)) / 360\).
Problem 74
The volume that has a base of the ellipse \(x^{2} / 4+y^{2} / 9=1\) and cross- sections of an equilateral triangle perpendicular to the \(y\) -axis. Use the method of slicing.
Problem 74
Find the antiderivative. \(\int x \ln \left(x^{2}\right) d x\)
Problem 75
Find the antiderivative. \(\int \frac{4 x^{2}}{\sqrt{1-x^{6}}} d x\)
Problem 76
\(x=y^{2}\) and \(x=3 y\) rotated around the \(y\) -axis using the washer method
