91Ó°ÊÓ

Find \(d y / d x\) $$ y=\int_{\tan x}^{0} \frac{d t}{1+t^{2}} $$

Find the areas of the regions enclosed by the lines and curves. $$ y=7-2 x^{2} \quad \text { and } \quad y=x^{2}+4 $$

Evaluate the integrals in Exercises \(17-50\) $$ \int x^{3} \sqrt{x^{2}+1} d x $$

Find the total area between the region and the \(x-\) axis. $$ y=-x^{2}-2 x, \quad-3 \leq x \leq 2 $$

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

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

Evaluate the integrals in Exercises \(17-50\) $$ \int 3 x^{5} \sqrt{x^{3}+1} d x $$

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

Find the total area between the region and the \(x-\) axis. $$ y=3 x^{2}-3, \quad-2 \leq x \leq 2 $$

Evaluate the integrals in Exercises \(17-50\) $$ \int \frac{x}{\left(x^{2}-4\right)^{3}} d x $$