## The Calculus Primer (2011)

### Part XIV. The Definite Integral

### Chapter 53. AREA UNDER A CURVE

**14—5. Calculation of Areas.** The discussion in §14—1 and §14—2 suggests a convenient method for calculating the area under a specific portion of a curve whose equation is known.

EXAMPLE 1. Find the area bounded by the parabola *y* = *x*^{2}*,* the *X*-axis, and the ordinates *x* = 1 and *x* = 3.

*Solution.*

EXAMPLE 2. Determine the area under the curve *y*^{2} = *x* and the line *x* = 9.

*Solution. y*^{2} *= x, or y = x*^{½}*.*

Area *ORS*

Hence

area *ORT =* (2) (18) = 36.

EXAMPLE 3. Find the area under the curve *y =* cos *x* from *x* = – π/2 to *x* = π/2.

*Solution.*

EXAMPLE 4. Find the area of the ellipse

*Solution.* We first find the area under this curve lying in the first quadrant.

From the given equation,

Since the shaded area lies entirely in the first quadrant, both *a* and *b* are positive. We may therefore disregard the ± sign. Hence

By formula [68], page 376:

Hence, area of entire ellipse equals 4*A*, or *πab,*

The reader should note carefully that this method of determining areas is a perfectly general method only if *y* = *f*(*x*) is a continuous, single-valued function for values of *x* between *x* = *a* and *x =* 6, and if the limits *a* and *b* are finite. If either of these conditions does not hold, the method may or may not yield correct results, and special methods must be used.

**EXERCISE 14—2**

**1.** Find the area bounded by the parabola *y* = 6*x*^{2}, the *X*-axis, and the ordinates at *x* = 2 and *x* = 4.

**2.** Find the area bounded by the parabola *y*^{2} = 4*x,* the *Y*-axis, and the line *y* = 4. *Hint*; First find the area *A* (the whole square); then find area *B* by difference.

**3.** Find the shaded area, given that the equation of the curve is *y* = *x*^{2} — 4.

**4.** Find the area under one arch of the curve *y* = sin *x,* from *x =* 0 to *x* = *π.*

**5.** Find by integration the area included by the *X*-axis and the lines *y* = *x +* 4 and 2*x* + *y* = 10; verify the result by elementary geometry.

**6.** Find the area *A* bounded by the curve *y* = *x*^{3}*,* the *X*-axis, and the line *x* = 2. Find also the unshaded area *B*.

**7.** Find the area of the figure bounded by the curve *y* = *x*^{2} *+* 2*x –* 6, the *X*-axis, and the ordinates at *x* = 2 and *x* = 6.

**8.** Find the area included between the hyperbola *xy* = *a*^{2}, the *X*-axis, and the ordinates at *x* = *a* and *x* = *b.*

**9.** Find the area bounded by the parabola *x*^{½} + *y*^{½} = *a*^{½} and the coordinate axes.

**10.** Determine the area lying between the two parabolas *y*^{2} = 8*x* and *x*^{2} = 8*y*.

**11.** Prove that the area bounded by a parabola and any chord drawn perpendicular to its axis is equal to two-thirds of the rectangle in which this area is inscribed.

**12.** Show by integration that the area of the circle *x*^{2} + *y*^{2} = *a*^{2} equals *πa*^{2}*;* first find the area in the first quadrant, using the integration formula

**14—6. Length of Arc.** The evaluation of a definite integral can also be used to determine the length of a particular arc of a curve whose equation is known. It will be recalled that

Now, consider the point *P*_{0} as a fixed point on the curve *y* = *f*(*x*)*,* and regard the length along the curve of an arc to be generated by a moving, variable point (*x,y*)*.* Reasoning as in §14—2, and integrating equation (1) above, we have:

remembering that the constant of integration is to be chosen so that *s* = 0 when *x* = *x*_{0}.

Thus,

Hence, by difference,

We shall discuss the length of arc further in Chapter Fifteen.