A large tank is designed with ends in the shape of the region between the curves $y=\frac{1}{2} x^2$ and $y=$ 12, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasoline's density is 42.0 lb/ft$^3$.)

A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Estimate the hydrostatic force on the top of the cube.

A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, estimate the hydro static force on the deep end.

A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Estimate the hydrostatic force on one of the sides of the cube.

A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, estimate the hydrostatic force on the bottom of the pool.

Find $P(X < 4)$.

After a 6-mg injection of dye into a heart, the readings of dye concentration at two-second intervals are as shown in the table. Use Simpson’s Rule to estimate the cardiac output.

$$\begin{array}{|r|l||c|l|} \hline{t} & c(t) & t & c(t) \\ \hline 0 & 0 & 14 & 4.7 \\ 2 & 1.9 & 16 & 3.3 \\ 4 & 3.3 & 18 & 2.1 \\ 6 & 5.1 & 20 & 1.1 \\ 8 & 7.6 & 22 & 0.5 \\ 10 & 7.1 & 24 & 0 \\ 12 & 5.8 & & \\ \hline \end{array}$$

Explain why the function

$$f(x) = \begin{cases}\dfrac{\pi}{20}\sin\left(\dfrac{\pi x}{10}\right) &\text{ if } 0 \leqslant x \leqslant 10 \\ 0 &\text{ if } x < 0 \text{ or } x > 10 \end{cases}$$

is a probability density function.

An anchor weighing 100 lb in water is attached to a chain weighing 3 lb/ft in water. Find the work done to haul the anchor and chain to the surface of the water from a depth of 25 ft.

https://admin.slader.com/textbook/9780357632499-calculus-concepts-and-contexts-5th-edition/534/exercises/38/

Use the parametric equations of an ellipse, x = a cos θ, y = b sin θ, 0 ≤ θ ≤ 2π, to find the area that it encloses.

A curve called Cornu’s spiral is defined by the parametric equations x = C(t) = ^t∫0 cos(πu^2/2) du, y = S(t) = ^t∫0 sin (πu^2/2) du, where C and S are the Fresnel functions that were introduced in Chapter 5. Find the length of Cornu’s spiral from the origin to the point with parameter value t.

If $f$ is a continuous function, what is the limit as $h \rightarrow 0$ of the average value of $f$ on the interval $[x, x + h]$?

A curve called Cornu’s spiral is defined by the parametric equations x = C(t) = ^t∫0 cos (πu^2/2)du, y = S(t) = ^t∫0 sin(πu^2/2)du, where C and S are the Fresnel functions. Graph this curve. What happens as t→∞ and as t→-∞?