Area in Rectangular Coordinates

Recall that the area under the graph of a continuous function \(f\left( x \right)\) between the vertical lines \(x = a,\) \(x = b\) can be computed by the definite integral:

where \(F\left( x \right)\) is any antiderivative of \(f\left( x \right).\)

We can extend the notion of the area under a curve and consider the area of the region between two curves.

If \(f\left( x \right)\) and \(g\left( x \right)\) are two continuous functions and \(f\left( x \right) \ge g\left( x \right)\) on the closed interval \(\left[ {a,b} \right],\) then the area between the curves \(y = f\left( x \right)\) and \(y = g\left( x \right)\) in this interval is given by

In terms of antiderivatives, the area of region is expressed in the form

where \(F\left( x \right)\) and \(G\left( x \right)\) are antiderivatives of the functions \(f\left( x \right)\) and \(g\left( x \right),\) respectively.

Note that this area will always be non-negative as \(f\left( x \right) - g\left( x \right) \ge 0\) for all \(x \in \left[ {a,b} \right].\)

If there are intersection points, we should break up the interval into several subintervals and determine which curve is greater on each subinterval. Then we can determine the area of each region by integrating the difference of the larger and the smaller function.

Area in Polar Coordinates

Consider the region \(OKM\) bounded by a polar curve \(r = f\left( \theta \right)\) and two semi-straight lines \(\theta =\alpha\) and \(\theta = \beta.\)

The area of the polar region is given by

The area of a region between two polar curves \(r = f\left( \theta \right)\) and \(r = g\left( \theta \right)\) in the sector \(\left[ {\alpha ,\beta } \right]\) is expressed by the integral

Area of a Region Bounded by a Parametric Curve

Recall that the area under a curve \(y = f\left( x \right)\) for \(f\left( x \right) \ge 0\) on the interval \(\left[ {a,b} \right]\) can be computed with the integral \(\int\limits_a^b {f\left( x \right)dx}.\) Suppose now that the curve is defined in parametric form by the equations

If the parameter \(t\) runs between \({t_1}\) and \({t_2}\) where

then the area under the curve is given by the formula

The functions \(x\left( t \right),\) \(x^\prime\left( t \right),\) \(y\left( t \right)\) here are assumed to be continuous on the interval \(\left[ {a,b} \right].\) Besides that, the function \(x\left( t \right),\) must be monotonic on this interval.

If \(x = x\left( t \right),\) \(y = y\left( t \right),\) \(0 \le t \le T\) are parametric equations of a smooth piecewise closed curve \(C\) traversed in the counterclockwise direction and bounding a region on the left (Figure \(5\)), then the area of the region is given by the following integrals:

Solved Problems

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Solution.

First we determine the points of intersection of the curves.

We set \(f\left( x \right) = g\left( x \right)\) to find the roots:

Solve the quadratic equation:

Thus, the cubic equation has two roots: \(x = -1\) (of multiplicity \(2\)) and \(x=2.\)

The area we wish to calculate is shown in Figure \(6\) below.

We can see from the Figure that the line \(y = 3x + 2\) lies above the cubic parabola \(y = {x^3}\) on the interval \(\left[ { - 1,2} \right].\) Hence, the area of the region is given by

Solution.

The area under the curve is given by the integral

Integrating yields:

Hence

Solution.

Equating both areas, we get the following:

Solution.

The area of the region is written in the form

Using the difference of sines identity

we obtain

The region has the largest area when \(\cos \left( {t + \frac{\pi }{4}} \right) = -1.\)

Solving this equation, we find