Chapter 6

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2}^{x^{2}-2} \sqrt{3+u} d u $$

We can approximate the monthly rainfall in Los Angeles using the formula \(p(t)=1.6+1.6 \cos 2 \pi t\), where \(t\) is the fraction of the year elapsed since January 1 and \(p(t)\) is the rainfall measured in inches/month. (a) Explain (without doing any calculations) that there will be a value of \(t\) for which the monthly rainfall is exactly equal to the annual average rainfall. (b) Find this value of \(t\) (there may be more than one). (c) Assuming there are 12 months in a year and each month has the same duration, show that the total rainfall in one year is: $$ P_{\text {total }}=12 \int_{0}^{1} p(t) d t $$ Explain in particular why the factor 12 is needed and what the units of \(P_{\text {total }}\) are. (d) Calculate \(P_{\text {total }}\) -

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{3 x}\left(1+e^{t}\right) d t $$

The daily temperature high in Minneapolis-St. Paul can be modeled using a formula: $$ T(t)=11.5+7.5 \cos (2 \pi(t-0.6)) $$ where \(T\) is measured in degrees Celsius and \(t\) measures the fraction of the year that has elapsed since January \(1 .\) (a) Find the average annual daily temperature high. (b) Find the time of year (value of \(t\) ) at which this average daily temperature high is actually observed.

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{2 x^{2}-1}\left(e^{-2 t}+e^{2}\right) d t $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}-1, y=x+1\)

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

In Problems 24-29, express the definite integrals as limits of Riemann sums. $$ \int_{-2}^{-1} \frac{x^{2}}{1+x^{2}} d x $$

Find the areas of the regions bounded by the lines and curves. \(y=e^{x / 2}, y=-x, x=0, x=2\)

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

In Problems , express the definite integrals as limits of Riemann sum $$ \int_{1}^{3}(x+1)^{1 / 3} d x $$

Find the areas of the regions bounded by the lines and curves. \(y=\cos x, y=0, x=0, x=\frac{\pi}{2}\)

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{x}^{5}\left(1+e^{t}\right) d t $$

In Problems , express the definite integrals as limits of Riemann sum $$ \int_{1}^{3} e^{-2 x} d x $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}+1, y=2 x, x=0\)

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

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=3 x-2\)

In Problems , express the definite integrals as limits of Riemann sum $$ \int_{0}^{\pi} \cos \frac{2 x}{\pi} d x $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=\frac{1}{x}, x=1, x=2\)

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{x}^{5} \frac{1}{u^{2}} d u, x>0 $$

In Problems , express the definite integrals as limits of Riemann sum $$ \int_{0}^{5} x^{3} d x $$

Find the areas of the regions bounded by the lines and curves. \(y=1, y=\cos x\) from \(x=0\) to \(x=\frac{\pi}{2}\)

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

In Problems 30-36, use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{0}^{5} e^{-x} d x $$

Find the areas of the regions bounded by the lines and curves. \(y=\sin x, y=1\) from \(x=0\) to \(x=\frac{\pi}{4}\)

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

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s. $$ \int_{-1}^{2}\left(x^{2}-1\right) d x $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=(x-2)^{2}, y=0\) from \(x=0\) to \(x=1\)

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

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s.$$ \int_{-2}^{2} \frac{1}{2} x^{3} d x $$

Find the areas of the regions bounded by the lines and curves. \(y=x^{2}, y=x^{3}\) from \(x=0\) to \(x=2\)

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

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s. $$ \int_{0}^{3}(2 x+1) d x $$

Find the areas of the regions bounded by the lines and curves. \(y=e^{-x}, y=x+1\) from \(x=-1\) to \(x=1\)

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{-x}^{x} u d u $$

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 \(y .\) \(y=x^{2}, y=(x-2)^{2}, y=0\) from \(x=0\) to \(x=2\)

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

In Problems , use a graph to interpret the definite integral in terms of areas. Do not compute the integrals.s. $$ \int_{-3}^{2}\left(1-\frac{1}{2} x\right) d x $$

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 \(y .\) \(y=x, y=x^{2}\)

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{x^{3}}^{x^{4}} \ln \left(1+t^{2}\right) d t $$

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. $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

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

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

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 \(h\).

Compute the indefinite integrals. $$ \int\left(x^{3}-4\right) d x $$