91Ó°ÊÓ

Find the areas of the regions bounded by the lines and curves by expressing \(x\) as a function of \(y\) and integrating with respect to \(y .\) \(y=x, y=0, y=1-x\), from \(x=0\) to \(x=1\)

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2-x^{2}}^{x+x^{3}}\left(t^{2}-1\right) d t $$

In Problems \(37-47\), use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{-2}^{3}|x| d x $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{1+x^{2}}^{x^{3}-2 x}(t+1) d t $$

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

In Problems 39-96, compute the indefinite integrals. $$ \int\left(1+3 x^{2}\right) d x $$

Find the volume of a right circular cone with base radius \(p\) and height \(h\).

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{2}^{5}\left(\frac{1}{2} x-4\right) d x $$

Find the volume of a pyramid with square base of side length \(a\) and height \(h\).

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{1 / 2}^{1} \sqrt{1-x^{2}} d x $$