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## Main Question or Discussion Point

Hi, the AP test is approaching and I'm struggling with these two Calculus questions and need some guidance. If you can provide a general (or detailed if possible =)) explanation on how you solved the problem, or the strategy you used for the problem, I would be extremely grateful. Solving even just one part will help me immensely! So here goes...

1. Let f and g be the functions given by f(x)=e^x and g(x) = ln(x)

a. Find the area of the region enclosed by the graphs of f and g between x = 1/2 and x=1.

b. Find the volume of the solid generated when the region enclosed by the graphs of f and g between x= 1/2 and x=1 is revolved around the line y=4.

c. Let h be the function given by h(x) = f(x) - g(x). Find the absolute minimum value of h(x) on the closed interval [1/2,1] and find the absolute maximum value of h(x) on the closed interval [1/2,1].

2.

t(minutes) 0 2 5 7 11 12

r ' (t) (feet/min) 5.7 4.0 2.0 1.2 0.6 0.5 (Note: this is a table)

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above gives seleceted values of the rate of change, r ' (t), of the radius of the balloon over the time interval [0,12]. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V = 4/3(pi)(r^3).

a. Find the rate of change of the volume of the balloon with respect to time when t=5. Indicate units of measure.

b. Use a right reimann sum with the five subintervals indicated by the data in the table to approximate the integral(antidifferentiation) from 0 to 12 of r ' (t) dt.

c. Using the correct units, explain the meaning of your solution from part b in terms of the radius of the balloon.

So there it is...any help is appreciated, good luck, and thanks in advance :X

1. Let f and g be the functions given by f(x)=e^x and g(x) = ln(x)

a. Find the area of the region enclosed by the graphs of f and g between x = 1/2 and x=1.

b. Find the volume of the solid generated when the region enclosed by the graphs of f and g between x= 1/2 and x=1 is revolved around the line y=4.

c. Let h be the function given by h(x) = f(x) - g(x). Find the absolute minimum value of h(x) on the closed interval [1/2,1] and find the absolute maximum value of h(x) on the closed interval [1/2,1].

2.

t(minutes) 0 2 5 7 11 12

r ' (t) (feet/min) 5.7 4.0 2.0 1.2 0.6 0.5 (Note: this is a table)

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above gives seleceted values of the rate of change, r ' (t), of the radius of the balloon over the time interval [0,12]. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V = 4/3(pi)(r^3).

a. Find the rate of change of the volume of the balloon with respect to time when t=5. Indicate units of measure.

b. Use a right reimann sum with the five subintervals indicated by the data in the table to approximate the integral(antidifferentiation) from 0 to 12 of r ' (t) dt.

c. Using the correct units, explain the meaning of your solution from part b in terms of the radius of the balloon.

So there it is...any help is appreciated, good luck, and thanks in advance :X