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Unit 8: Applications of Integration

13 dot points across 13 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you find the length of a smooth curve using a definite integral?

Topic 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled: compute the length of a curve y = f(x) and the distance a particle travels using the arc length integral (BC).
A focused answer to AP Calculus BC Topic 8.13, computing the arc length of a smooth planar curve with the definite-integral formula and using it for distance traveled, with worked examples.
9 min answer →

How do you find the area between two curves given as functions of x?

Topic 8.4 Finding the Area Between Curves Expressed as Functions of x: integrate the top minus the bottom curve to find the enclosed area.
A focused answer to AP Calculus AB Topic 8.4, finding the area between two curves given as functions of x by integrating the upper minus the lower function between intersection points, with worked examples.
9 min answer →

When is it easier to find an area by integrating with respect to y, and how do you do it?

Topic 8.5 Finding the Area Between Curves Expressed as Functions of y: integrate right minus left with respect to y to find the enclosed area.
A focused answer to AP Calculus AB Topic 8.5, finding the area between curves by integrating right minus left with respect to y, when this avoids splitting the region, with worked examples.
9 min answer →

How do you find the area between curves that cross more than once and swap which is on top?

Topic 8.6 Finding the Area Between Curves That Intersect at More Than Two Points: split the area at each crossing where the top and bottom curves swap.
A focused answer to AP Calculus AB Topic 8.6, finding the area between curves that intersect more than twice by splitting the integral where the upper and lower curves swap, with worked examples.
9 min answer →

How do you recover velocity and position from acceleration, and find displacement and total distance, using integrals?

Topic 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals: use integrals to find velocity, position, displacement and total distance.
A focused answer to AP Calculus AB Topic 8.2, using integrals to recover velocity and position from acceleration and to compute displacement and total distance travelled, distinguishing the two, with worked examples.
10 min answer →

How do you find the average value of a continuous function over an interval using a definite integral?

Topic 8.1 Finding the Average Value of a Function on an Interval: compute the average value of a function with the definite-integral formula.
A focused answer to AP Calculus AB Topic 8.1, computing the average value of a continuous function over an interval with the integral formula, distinguishing it from the average rate of change, with worked examples.
8 min answer →

How do you use a definite integral of a rate to find the net change in an applied quantity?

Topic 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts: find net change in a quantity by integrating its rate in context.
A focused answer to AP Calculus AB Topic 8.3, using definite integrals of rates to find net change in applied quantities such as water in a tank, with the starting-amount-plus-net-change structure and worked examples.
10 min answer →

How do you find the volume of a solid of revolution using the disc method about the x- or y-axis?

Topic 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis: find the volume of a solid of revolution about a coordinate axis using the disc method.
A focused answer to AP Calculus AB Topic 8.9, finding volumes of solids of revolution about the x- or y-axis with the disc method, integrating pi times the radius squared, with worked examples.
9 min answer →

How do you find the volume of a solid of revolution about a horizontal or vertical line other than a coordinate axis using the disc method?

Topic 8.10 Volume with Disc Method: Revolving Around Other Axes: find the volume of a solid of revolution about a line other than a coordinate axis using the disc method.
A focused answer to AP Calculus AB Topic 8.10, finding volumes of solids of revolution about lines other than the coordinate axes with the disc method by adjusting the radius for the shifted axis, with worked examples.
9 min answer →

How do you find the volume of a solid of revolution with a hole using the washer method about the x- or y-axis?

Topic 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis: find the volume of a solid of revolution with a hole about a coordinate axis using the washer method.
A focused answer to AP Calculus AB Topic 8.11, finding volumes of solids of revolution about the x- or y-axis with the washer method, integrating pi times outer radius squared minus inner radius squared, with worked examples.
10 min answer →

How do you find the volume of a solid of revolution with a hole about a line other than a coordinate axis using the washer method?

Topic 8.12 Volume with Washer Method: Revolving Around Other Axes: find the volume of a solid of revolution with a hole about a line other than a coordinate axis using the washer method.
A focused answer to AP Calculus AB Topic 8.12, finding volumes of solids of revolution about lines other than the coordinate axes with the washer method by shifting both radii, with worked examples.
10 min answer →

How do you find the volume of a solid with square or rectangular cross sections built on a region?

Topic 8.7 Volumes with Cross Sections: Squares and Rectangles: integrate the cross-sectional area to find volume when cross sections are squares or rectangles.
A focused answer to AP Calculus AB Topic 8.7, finding volumes of solids with square or rectangular cross sections perpendicular to an axis by integrating the cross-sectional area, with worked examples.
9 min answer →

How do you find the volume of a solid with triangular or semicircular cross sections built on a region?

Topic 8.8 Volumes with Cross Sections: Triangles and Semicircles: integrate the cross-sectional area to find volume when cross sections are triangles or semicircles.
A focused answer to AP Calculus AB Topic 8.8, finding volumes of solids with triangular or semicircular cross sections by integrating the cross-sectional area, with the correct area formulas and worked examples.
9 min answer →