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Application of Integrals Notes




Area under a simple curve

  • Here we need to find the the area bounded by the curve y = f (x), x-axis and the ordinates x = a and x = b
  • we can think of area under the curve as composed of large number of very thin vertical strips.
  • Consider an arbitrary strip of height y and width dx, then dA (area of the elementary strip)= ydx, where, y = f(x).

  • Total area between x=a and x=b can be obtained by adding up the elementary areas of thin strips across the region
  • A= $\int_{a}^{b} f(x) \; dx$
  • Similary The area A of the region bounded by the curve x = g (y), y-axis and the lines y = a,y = b is given as

  • A= $\int_{a}^{b} g(y) \; dy$

Some Important points

If the position of the graph under consideration is below x axis then the the integration will come out negative but since we are talking about area. we can take the positive value

A= |$\int_{a}^{b} f(x) \; dx$ |
If the position of the graph under consideration is below x axis for some part and above x axis for some part.then we need to calculate the integration of both the parts separately and then just add the positive value

A= $\int_{a}^{c} f(x) \; dx$ + |$\int_{c}^{b} f(x) \; dx|$

Area of the region bounded by a curve and a line

Area of the region bounded by a curve and a line can be found in the same way as explained above
Lets take few example
(1) Find the area of the region bounded by the curve $y = x^2$ and the line y = 4.

Solution
(a)First we need to find the points where the curve and the line intersect. To do this, set the equation of the curve equal to the equation of the line and solve it
So, here $4=x^2$ or $x=\pm 2$, So point of intersection are (-2.4) and (2,4)
(b)we also see that that Area around y axis is symmetrical, so we can find on one side and double it
(c) Consider a vertical strip dx, The length of the strip will be Y coordinate of line - Y coordinate of curve

So Area of bounded region
$A=2\int_{0}^{2} (4-x^2) \; dx= \frac {32}{3}$

Area between Two Curves

(i)The area of the region enclosed between two curves y = f (x), y = g (x) andthe lines x = a, x = b is given by the formula

$A= \int_{a}^{b} [f(x) -g(x)] \; dx$ , where, $f (x)\geq g (x)$ in [a, b]

(ii) If $f (x) \geq g (x)$ in [a, c] and $f (x) \leq g (x)$ in [c, b], a < c < b, then

$A= \int_{a}^{c} [f(x) -g(x)] \; dx + \int_{c}^{b} [g(x) -f(x)] \; dx$

How to find Area between two curves

1. Sketch a rough plot of the given curve and line on the same set of axes. This will give you a visual idea of the region whose area you wish to find.
2. Find the points where the curve and the line intersect. To do this, set the equation of the curve equal to the equation of the line and solve for \(x\) (or \(y\) if it's more suitable).
3. After plotting, determine which function is the top function and which one is the bottom for the interval you are interested in. You will subtract the bottom function from the top one to find the height of the thin strip (or differential element) that you'll use in integration.
4.To find the area, integrate as follows:
\[ \text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx \]
Where:
- \(f(x)\) is the top function
- \(g(x)\) is the bottom function
- \(a\) and \(b\) are the x-values of the points of intersection found in step 2.
5. Compute the integral to get the area.

Solved Examples

Example 1
Find the area of the region bounded by the curve \(y = x^2\) and the line \(y = 2x\).
Solution

1. Plot them. You'll see they intersect in two places.
2. Find points of intersection:
Setting \(x^2 = 2x\), we get \(x^2 - 2x = 0 \implies x(x-2) = 0\). So, the points are \(x = 0\) and \(x = 2\).
3. From the graph, between \(x = 0\) and \(x = 2\), \(2x\) is above \(x^2\).
4. Setup the integral:
\[ \text{Area} = \int_{0}^{2} [2x - x^2] \, dx \]
\[ = \left[ x^2 - \frac{x^3}{3} \right]_0^2 \]
\[ = \left[4 - \frac{8}{3}\right] \]
\[ = \frac{4}{3} \]
So, the area between the curve and the line in this interval is \( \frac{4}{3} \) square units.

Example 2
Find the area between the curves \(y = \sqrt{x}\) and \(y = x^2\).
Solution

1. Plot them. You'll see they intersect in two places.
Set \(x^2 = \sqrt{x}\).
\[x^4 = x\]
\[x^4 - x = 0\]
\[x(x^3 - 1) = 0\]
This gives \(x = 0\) and \(x = 1\) as the points of intersection.
2. In this interval, \(y = \sqrt{x}\) is above \(y = x^2\).
3. Calculate the area
\[ \text{Area} = \int_{0}^{1} (\sqrt{x} - x^2) \, dx \]
\[ = \left[ \frac{2x^{3/2}}{3} - \frac{x^3}{3} \right]_0^1 \]
\[ = \frac{2}{3} - \frac{1}{3} \]
\[ = \frac{1}{3} \]

Example 3
Find the area bounded by the curve y = cos x between x = 0 and $x = 2 \pi$.
Solution
First plot the curve

A= $\int_{0}^{\pi/2} cos x \; dx$ + |$\int_{\pi/2}^{3\pi/2} cos x \; dx$| + $\int_{3\pi/2}^{2\pi} cos x \; dx$
A=4




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