91Ó°ÊÓ

Find \(d y / d x\).$$y=\int_{1}^{x} \frac{1}{t} d t, \quad x > 0.$$

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 \([a, b]\). \(f(x)=2 x\) over the interval [0,3].

Find \(d y / d x\).$$y=\int_{\sqrt{x}}^{0} \sin \left(t^{2}\right) d t.$$

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

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 \([a, b]\). \(f(x)=x^{2}+1\) over the interval [0,3].

Evaluate the integrals. $$\int \sqrt{\frac{x^{3}-3}{x^{11}}} d x$$

Find \(d y / d x\).$$y=x \int_{2}^{x^{2}} \sin \left(t^{3}\right) d t.$$

Evaluate the integrals. $$\int \sqrt{\frac{x^{4}}{x^{3}-1}} d x$$

Find the areas of the regions enclosed by the lines and curves. $$y=2 x-x^{2} \quad \text { and } \quad y=-3$$

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 \([a, b]\). \(f(x)=3 x^{2}\) over the interval [0,1].