Application of Integrals

Integrals are used to find the area and volume of various 2-D and 3-D curves, and they have vast applications in the fields of mathematics and physics. They generally help us to calculate the area of the curve, irregular contour, the volume of various curves, and others.

Applications of Integrals are shown below in the figure. Integrals are used to find areas under curves, areas between curves, and centroids of regions.

Find the Area Between a Curve and an Axis

We can find the Area between the curve y = f(x), the x-axis, and specific intervals that are the lines x = a and x = b by using integration:

∫_a^b y dx = ∫_a^b f(x) dx = F(b) -F(a)

Similarly, when dealing with the region enclosed by the curve x= g(y), the y-axis, and the lines y = a and y = b, the Integral expression is:

∫_a^b x.dy = ∫_a^b g(y) dy = G(b) - G(a)

Find the Area Between Two Curves

For areas between two curves y = g(x) and y = f(x), where f(x) ≥ g(x) in the interval [a, b], the area between x = a and x = b is:

∫_a^b f(x).dx - ∫_a^b g(x).dx = ∫_a^b {f(x) - g(x)}.dx

Similarly, for regions between two curves x = g(y) and x = f(y), where f(y) ≥ g(y) in the interval [c, d], the Integral expression becomes:

∫_a^b f(y).dy - ∫_a^b g(y).dy = ∫_a^b {f(y) - g(y)}.dy

Find Area Under Curve

To calculate the area under a curve, follow the steps added below

Step 1: Firstly, identify the equation of the curve y = f(x), the limits, and the axis for area calculation.
Step 2: The integration (antiderivative) of the curve is found.
Step 3: The upper and lower limits are applied to the integral result, and the difference gives the area under the curve.

Area = ∫_a^b y.dx

⇒ Area = ∫_a^b f(x).dx

⇒ Area = [g(x)]^b_a

Area = g(b) − g(a)