Definite Integrals Solved Examples (Class 12 Mathematics)
Definite integrals numericals develop integration skills with applications to areas and volumes. These problems form the foundation for applied mathematics
TL;DR: Definite integrals numericals develop integration skills with applications to areas and volumes. These problems form the foundation for applied mathem…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
Definite integrals numericals develop integration skills with applications to areas and volumes. These problems form the foundation for applied mathematics
Definite Integrals — Solved Numerical Examples (Step by Step)
Example 1: Evaluate the definite integral ∫(0 to 2) (3x² + 2x) dx.
Solution: ∫(3x² + 2x) dx = x³ + x². Evaluating from 0 to 2: [x³ + x²] from 0 to 2 = (8 + 4) - (0 + 0) = 12.
Example 2: Evaluate ∫(1 to 3) (1/x) dx.
Solution: ∫(1/x) dx = ln|x|. Evaluating from 1 to 3: [ln|x|] from 1 to 3 = ln(3) - ln(1) = ln(3) - 0 = ln(3) ≈ 1.099.
Example 3: Find the area under the curve y = x² from x = 0 to x = 2.
Solution: Area = ∫(0 to 2) x² dx = [x³/3] from 0 to 2 = (8/3) - 0 = 8/3 ≈ 2.667 square units.
Example 4: Evaluate ∫(0 to π/2) sin(x) dx.
Solution: ∫sin(x) dx = -cos(x). Evaluating from 0 to π/2: [-cos(x)] from 0 to π/2 = -cos(π/2) - (-cos(0)) = 0 + 1 = 1.
Example 5: Evaluate ∫(0 to 1) e^x dx.
Solution: ∫e^x dx = e^x. Evaluating from 0 to 1: [e^x] from 0 to 1 = e - 1 ≈ 1.718.
Tips
- Fundamental theorem of calculus: ∫(a to b) f(x) dx = F(b) - F(a) where F is antiderivative of f.
- Common integrals: ∫x^n dx = x^(n+1)/(n+1), ∫e^x dx = e^x, ∫sin(x) dx = -cos(x).
- Definite integral gives signed area: regions below x-axis count as negative.
Frequently Asked Questions
What is the difference between indefinite and definite integrals?
Indefinite integral ∫f(x) dx gives a family of antiderivatives with constant C. Definite integral ∫(a to b) f(x) dx gives a specific numerical value (the area).
Why do we use limits in definite integrals?
Limits define the interval over which we calculate the area. The definite integral finds the net signed area between the curve and x-axis from x=a to x=b.
More Mathematics Solved Examples
- Real Numbers (HCF LCM Euclid)
- Pair of Linear Equations in Two Variables
- Circles
- Statistics
- Linear Equations in Two Variables
- Heron's Formula
🤖 Stuck on any of these? Ask Syllab's free AI Tutor to explain step by step →