Let $L$ be the lamina of uniform density $\rho=1$ obtained by removing circle $A$ of radius $r$ from circle $B$ of radius $2 r$ (see figure).

What is the center of mass of $L$?

A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.33 m/s at an angle of $30 ^ { \circ }$ with respect to the original line of motion. Was the collision inelastic or elastic?

a billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass,. Calling $(A)$ the ball that was originally moving and $(B)$ the ball that was originally stationary, and considering that, after the collision, ball $B$ ends up moving at $40\,\%$ the starting speed of ball $A$, what is the speed of ball A after the collision?

An object of mass 3.00 kg, moving with an initial velocity of m/s, collides with and sticks to an object of mass 2.00 kg with an initial velocity of m/s. Find the final velocity of the composite object. $5.00 \hat { \mathrm { i } }$ $- 3.00 \hat { \mathrm { j } }$

A 1200-kg car traveling initially at $v_{\mathrm{C} i}=25.0 \mathrm{m} / \mathrm{s}$ in an easterly direction crashes into the back of a 9000-kg truck moving in the same direction at $v_{\mathrm{T} i}=20.0 \mathrm{m} / \mathrm{s}$. The velocity of the car immediately after the collision is $v_{C f}=18.0 \mathrm{m} / \mathrm{s}$ to the east. (a) What is the velocity of the truck immediately after the collision? (b) What is the change in mechanical energy of the car-truck system in the collision? (c) Account for this change in mechanical energy.

A baseball approaches home plate at a speed of 45.0 m/s, moving horizontally just before being hit by a bat. The batter hits a pop-up such that after hitting the bat, the baseball is moving at 55.0 m/s straight up. The ball has a mass of 145 g and is in contact with the bat for 2.00 ms. What is the average vector force the ball exerts on the bat during their interaction?

Find M_x, M_y, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. x = y + 2, x = y²

Identify the precalculus formula and representative element used to develop the integration formula for each application.

Area of a surface of revolution

Let $L$ be the lamina of uniform density $\rho=1$ obtained by removing circle $A$ of radius $r$ from circle $B$ of radius $2 r$ (see figure).

Find $M_y$ for $B$ and $M_y$ for $A$. Then use the previous part to compute $M_y$ for $L$.

Let $L$ be the lamina of uniform density $\rho=1$ obtained by removing circle $A$ of radius $r$ from circle $B$ of radius $2 r$ (see figure).

Show that $M_y$ for $L$ is equal to $\left(M_y\right.$ for $\left.B\right)-\left(M_y\right.$ for $\left.A\right)$.

Approximate the arc length of the graph of the function over the interval [0, 4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when x=0, x=1, x=2, x=3, and x=4, Find the sum of the four lengths. (c) Use Simpson’s Rule with n=10 to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated arc length. f(x) = x³

Identify the precalculus formula and representative element used to develop the integration formula for each application.

Shell method

Use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines.

$$y=x^3, \quad y=0, \quad x=2$$

x=4

Identify the precalculus formula and representative element used to develop the integration formula for each application.

Washer method