Exercise practice: Application of Integrals

3.1 How the exercises are arranged

The chapter has two exercise sets and a miscellaneous exercise. Exercise 8.1 asks for the area under a simple curve, usually a line, a parabola, a circle or an ellipse, with the region given by the curve and one or two lines. Exercise 8.2 asks for the area between two curves or between a curve and a line, where the curves must be solved together to find the limits. The miscellaneous exercise mixes both and adds problems where the area is given and a parameter has to be found.

3.2 Exercise 8.1 patterns

The standard pattern is to identify the curve, find the x-limits from the lines given, decide the upper and lower boundary and integrate. For the parabola y squared equal to x between x equal to 0 and x equal to 1, the upper branch is root x and the area is the integral from 0 to 1, giving two thirds. For the ellipse given between two vertical lines, the area is twice the upper half between zero and the line, and the answer contains an inverse sine term. For a region bounded by a line and the axes, the integral of the line gives the triangle area, and the answer should agree with the elementary formula.

3.3 Exercise 8.2 patterns

The curves must meet, so solve them simultaneously. For y equal to x squared and y equal to x, the meeting points are zero and one. Decide which is upper by testing one interior point, then integrate the difference. For y squared equal to four x and x equal to 4 a, a parameter appears and the final answer is a multiple of a squared. For two parabolas with axes perpendicular, solving gives the meeting points, and symmetry halves the work.

3.4 The miscellaneous exercise

This set adds the parameter problems: the area is given, the boundary contains a parameter and the parameter is found by equating the integral to the given value. It also adds regions defined by inequalities such as a circle cut by a line, where the sketch decides whether the wanted piece is the minor segment or the major part. The last exercises combine both curves and both limits, so the four-step method from the topic page is used in full.

3.5 A worked mixed exercise

A typical miscellaneous question gives the region bounded by the parabola y equal to x squared and the line y equal to x plus two. Solve x squared equal to x plus two to get the meeting points at x equal to minus one and x equal to two. Between these points the line is above the parabola, so the area is the integral of x plus two minus x squared from minus one to two. The antiderivative is x squared over two plus two x minus x cubed over three. At the upper limit two it is ten thirds, at the lower limit minus one it is minus seven sixths, and the difference gives nine halves square units. The sketch confirms that the line crosses the parabola at the two limits and lies above it between them.

3.6 A worked parameter question

A question may give the area and ask for a parameter in the boundary. For the region bounded by the line y equal to m x, the circle x squared plus y squared equal to four and the x-axis in the first quadrant, the area is a sector of the circle and equals twice the angle the line makes with the axis. Setting this equal to the given value gives the angle, and the tangent of that angle is m. The method is always the same: integrate first with the parameter left as a symbol, then equate to the given area and solve the resulting equation.

3.7 Common mistakes in the exercises

The most common errors are using the meeting points of the two curves as limits without checking the region asked for, forgetting that a curve below the axis contributes a negative value to the signed integral, and reversing the upper and lower functions. Tracing the region on the sketch and labelling which curve is upper on each piece prevents all three. A final check on the sign of the answer is also useful: an area can never be negative.

3.8 A final check on the exercises

Every exercise answer should end with square units and a sentence naming the region. If the integral of a difference came out negative, the upper and lower labels were reversed. If the answer is twice too large, a symmetric doubling was applied twice. These two checks catch almost every error in the exercise set and take less time than restarting the calculation. A good final check is to read the answer back in words: the area of this region is so many square units. If the sentence sounds wrong, the setup or the limits are wrong.