2006 AP Calculus Free Response Question 1 Aimee Davis and Sarah Laubach

The Question Let R be the region bounded by the graph of y=lnx and the line y=x-2, as shown above Let R be the region bounded by the graph of y=lnx and the line y=x-2, as shown above

The Parts A: Find the area of R A: Find the area of R B: Find the volume of the solid generated when R is rotated about the horizontal line y=-3 B: Find the volume of the solid generated when R is rotated about the horizontal line y=-3 C: Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the Y-axis C: Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the Y-axis

Part A To find the area of the shaded region “R” you must first take the integral of the upper curve minus the lower curve To find the area of the shaded region “R” you must first take the integral of the upper curve minus the lower curve Look at the graph to discern which graph is the upper curve and which is the lower curve Look at the graph to discern which graph is the upper curve and which is the lower curve In this case, it can be seen that y=lnx is the upper curve and y=x-2 is the lower curve… In this case, it can be seen that y=lnx is the upper curve and y=x-2 is the lower curve…

Lower Curve y=x-2 Upper Curve y=lnx

The Integral Used The equation used is: The equation used is: It can be simplified to: It can be simplified to:

Finding Points of Intersection (POI) The points of intersection for the two graphs are the same as the upper and lower limits for the area The points of intersection for the two graphs are the same as the upper and lower limits for the area Make sure the functions are typed into the y= screen Make sure the functions are typed into the y= screen Press: Press: 2 nd 2 nd CALC CALC 5: Intersect 5: Intersect For Left POI: For Left POI: On the upper curve, scroll to a point below the left POI On the upper curve, scroll to a point below the left POI Press ENTER Press ENTER On the lower curve, scroll to a point below the left POI On the lower curve, scroll to a point below the left POI Press ENTER Press ENTER Left POI will show up on bottom left side of calculator screen Left POI will show up on bottom left side of calculator screen

Finding the POI To find the Right POI: Repeat the steps necessary in finding the left POI, but scroll above the point on the graphs Right POI will show up on bottom left side of calculator screen

To Solve Part A Clear the equations out of the y= screen Clear the equations out of the y= screen Enter the simplified integral y=lnx-x+2 into the y= screen Enter the simplified integral y=lnx-x+2 into the y= screen Press: Press: 2 nd 2 nd CALC CALC Enter the upper and lower limits (the POIs) Enter the upper and lower limits (the POIs) The answer will appear on the bottom left hand side of the calculator screen The answer will appear on the bottom left hand side of the calculator screen

Part A Answer

Part B To find the volume of the solid generated when R is rotated about the horizontal line y=-3: To find the volume of the solid generated when R is rotated about the horizontal line y=-3: Use the Washer formula: Use the Washer formula: To form the integral, insert the outer function for R(x) and the inner function as r(x) To form the integral, insert the outer function for R(x) and the inner function as r(x)

Write the Integral Type the integral formed into the y= screen on the calculator so it looks like this: Type the integral formed into the y= screen on the calculator so it looks like this: The added +3 comes from adding the distance between the functions and the line y=-3 The added +3 comes from adding the distance between the functions and the line y=-3

Calculations With the equation in the y= screen click Graph 2nd Calc Enter in the limits ENTER

ANSWER!

Part C Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis

Writing the Integral Because the solid desired is now being rotated around the y-axis the outer and inner integrals are reversed, but the washer formula remains the same, the final integral should look like this: Because the solid desired is now being rotated around the y-axis the outer and inner integrals are reversed, but the washer formula remains the same, the final integral should look like this:

It’s the End as We Know It!