Applications of Integral Calculus
Integral calculus has different applications. Some of them are:
- To find the area under the curve.
- To find the area between two curves.
- To find the volumes.
- To find displacement and motion.
Example 1: Find the area of the region bounded by the curve y2 = 16x, x = 1, x=3 and the x-axis in first quadrant.
Solution:
First, we draw the figure of the curve y2 = 16x, x = 1, x=3 and the x-axis in first quadrant
y2 = 16x
⇒ y = ±√(16x)
⇒ y = ±4√x . . . (1)
We will take positive value of y as we have to find area under the first quadrant.
The required area is the shaded region PQRS.
Required area = [Tex]\int\limits^3_1 [/Tex]y dx
⇒ Required area = \int\limits^3_1 4√xdx [From equation 1]
⇒ Required area = 4[x½+1 / {(1/2)+1}]13
⇒ Required area = 4[x3/2 / (3/2)]13
⇒ Required area = 4[(2x3/2)/3]13
⇒ Required area = (8/3) [ 33/2 – 13/2]
⇒ Required area = (8/3) [3√3 – 1]
⇒ Required area = [8√3 – (8/3)] sq units
Example 2: Find the volume of solid bounded between the region x2 +y2 ≤ 4 and 0 ≤ z ≤ 1.
Solution:
From the given equation radius of disc = 2 units.
Required Volume = ∭V dx dy dz
⇒ Required volume = ∬R[Tex]\int\limits_{z=0}^{z=1} [/Tex](dz)dx dy
⇒ Required volume = ∬R[z]01dxdy = ∬R[1 – 0]dx dy
⇒ Required volume = ∬R[z]01dxdy = ∬R (1)dx dy
⇒ Required volume = ∬R dx dy
Since, ∬Rdx dy = Area (given shape is a disc which is a circle and area = πr2 )
⇒ Required volume = π(2)2
⇒ Required volume = 4π cubic units
Example 3: Find the displacement of the particle over the interval [1, 3] if the velocity of the particle is given by v(t) = 3t2 + 2t.
Solution:
Given the velocity v(t) = 3t2 + 2t and the interval [1, 3]
Displacement S(t) = [Tex]\int\limits_a^b [/Tex]v(t) dt
Here, a = 1 and b = 3 by the given interval
⇒ S(t) = [Tex]\int\limits_1^3 [/Tex](3t2 + 2t)dt
⇒ S(t) = [Tex]\int\limits_1^3 [/Tex]3t2dt + [Tex]\int\limits_1^3 [/Tex]2tdt
⇒ S(t) = 3[(t3/3)]13 + 2[(t2/ 2)]13
⇒ S(t) = [t3]13 + [t2]13
⇒ S(t) = [33 – 13] + [32 – 12]
⇒ S(t) = [27 – 1] + [ 9 – 1]
⇒ S(t) = 26 + 8
⇒ S(t) = 34 units
The displacement is 34 units.
Integral Calculus
Integral Calculus is the branch of calculus that deals with topics related to integration. Integrals are major components of calculus and are very useful in solving various problems based on real life. Some of such problems are the Basel problem, the problem of squaring the circle, the Gaussian integral, etc. Integral Calculus is directly related to differential calculus.
This article is a brief introduction to Integral Calculus, including topics such as fundamental theorems of integral calculus, types of integral, and integral calculus formulas, definite and indefinite integrals with their properties, applications of integral calculus, and their examples.
Table of Content
- What is Integral Calculus?
- Fundamental Theorems of Integral Calculus
- Integral Definition
- Types of Integrals
- Definite Integrals
- Definite Integral Formula
- Properties of Definite Integrals
- Indefinite Integrals
- Properties of Indefinite Integrals
- Improper Integrals
- Multiple Integrals
- Integral Calculus Formulas
- Methods to Find Integrals
- Applications of Integral Calculus
- Differential vs Integral Calculus
- Integral Calculus Examples
- Practice Problems on Integral Calculus