91Ó°ÊÓ

Evaluate the integrals. $$\int x(x-1)^{10} d x$$

Find the areas of the regions enclosed by the lines and curves. $$y=x^{4} \quad \text { and } \quad y=8 x$$

Find \(d y / d x\).$$y=\int_{-1}^{x} \frac{t^{2}}{t^{2}+4} d t-\int_{3}^{x} \frac{t^{2}}{t^{2}+4} d t.$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=x+x^{2}\) over the interval [0,1].

Find the areas of the regions enclosed by the lines and curves. $$y=x^{2}-2 x \text { and } y=x$$

Find \(d y / d x\).$$y=\left(\int_{0}^{x}\left(t^{3}+1\right)^{10} d t\right)^{3}.$$

Evaluate the integrals. $$\int x \sqrt{4-x} d x$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=3 x+2 x^{2}\) over the interval [0,1].

Find the areas of the regions enclosed by the lines and curves. $$y=x^{2} \quad \text { and } \quad y=-x^{2}+4 x$$

Find a formula for the Riemann sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals and using the right-hand endpoint for each \(c_{k} .\) Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). \(f(x)=2 x^{3}\) over the interval [0,1].