Integral Calculus Problem Example 3

Description:

Problem

Find the area bounded by the curve y = e^-x, the X − axis and the Y − axis.

Solution

Plot of y = e^-x, looks like following −

• The curve in magenta represents, y = e^-x.
• The area shaded in green represents the area bounded by the curve, X-axis and Y-axis.

Now,

For any function, y = f(x), Area under the curve between x = a and x = b is −

b∫a f(x) . dx

In our problem,

• f(x) = e^-x
• a = 0
• b = ∞,

Hence,

Area = ∞∫0 e^-x . dx

Substitution, t = -x, Hence, dt = -dx

Area = -∞∫0 -e^t . dt = -e^t]^-∞₀ = -[e^-∞ - e⁰] = -[0 - 1] = 1

Note

If the graph of y = e^-x, is unknown to a student, he/she may still be able to deduce the correct limits of integration. Following steps should be followed for the same −

• The problem asks us to find the area bounded by the curve, y = e^-x, the X − axis and the Y − axis.

• We know the limits of integration are the two values of ‘x’.

• Therefore, the Y − axis intersects the X − axis only at x = 0. Hence, one of the limits is certainly x = 0.

• The curve, y = e^-x, will intersect X − axis at y = 0 or e^-x = 0. Therefore, in this case, x → ∞.

Hence, the limits will be from x = 0 to x → ∞.