5.4 Integration formulas and the net change theorem  (Page 5/8)

Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral.

Write an integral that quantifies the increase in the volume of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral.

Suppose that a particle moves along a straight line with velocity $v\left(t\right)=4-2t,$ where $0\le t\le 2$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=2.$

Suppose that a particle moves along a straight line with velocity defined by $v\left(t\right)=t^2-3t-18,$ where $0\le t\le 6$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=6.$

Suppose that a particle moves along a straight line with velocity defined by $v\left(t\right)=\left|2t-6\right|,$ where $0\le t\le 6$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=6.$

Suppose that a particle moves along a straight line with acceleration defined by $a\left(t\right)=t-3,$ where $0\le t\le 6$ (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to $t=6$ if $v\left(0\right)=3$ and $d\left(0\right)=0.$

A ball is thrown upward from a height of 1.5 m at an initial speed of 40 m/sec. Acceleration resulting from gravity is −9.8 m/sec ² . Neglecting air resistance, solve for the velocity $v\left(t\right)$ and the height $h\left(t\right)$ of the ball t seconds after it is thrown and before it returns to the ground.

A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is −9.8 m/sec ² . Neglecting air resistance, solve for the velocity $v\left(t\right)$ and the height $h\left(t\right)$ of the ball t seconds after it is thrown and before it returns to the ground.

The area $A\left(t\right)$ of a circular shape is growing at a constant rate. If the area increases from 4 π units to 9 π units between times $t=2$ and $t=3,$ find the net change in the radius during that time.

A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from 36 π in. ³ to 288 π in. ³ between time $t=30$ and $t=60$ seconds, find the net change in the radius of the balloon during that time.

Water flows into a conical tank with cross-sectional area πx ² at height x and volume $\frac{\pi x^3}{3}$ up to height x . If water flows into the tank at a rate of 1 m ³ /min, find the height of water in the tank after 5 min. Find the change in height between 5 min and 10 min.

A horizontal cylindrical tank has cross-sectional area $A\left(x\right)=4\left(6x-x^2\right)m^2$ at height x meters above the bottom when $x\le 3.$

1. The volume V between heights a and b is ${\displaystyle {\int }_a^{b}A\left(x\right)dx}.$ Find the volume at heights between 2 m and 3 m.
2. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, $\frac{dx}{dt}=\frac{dx}{dV}\frac{dV}{dt},$ at how many meters per minute is the height of oil in the tank changing, expressed in terms of x , when the height is at x meters?
3. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?