Another Simple Example of Double Integral Evaluation
We aim to evaluate the following double integral:
∫₀¹ ∫₀¹ (x + y²) dy dx
Step 1: Integrate with respect to \( y \)
First, we focus on the inner integral, which is with respect to \( y \):
∫₀¹ (x + y²) dy
This simplifies to:
x[y]₀¹ + [y³/3]₀¹ = x(1 – 0) + (1/3 – 0) = x + 1/3
Step 2: Integrate with respect to \( x \)
Next, we integrate the resulting expression with respect to \( x \):
∫₀¹ (x + 1/3) dx
Applying the power rule for integration, we get:
[x²/2]₀¹ + [x/3]₀¹ = 1/2 + 1/3 = 5/6
Final Result
The value of the double integral is \( 5/6 \).
Yet Another Simple Example of Double Integral Evaluation
We aim to evaluate the following double integral:
∫₀² ∫₀¹ (2x + 3y) dy dx
Step 1: Integrate with respect to \( y \)
First, we tackle the inner integral, which is with respect to \( y \):
∫₀¹ (2x + 3y) dy
This simplifies to:
2x[y]₀¹ + 3[y²/2]₀¹ = 2x(1 – 0) + 3(1/2 – 0) = 2x + 3/2
Step 2: Integrate with respect to \( x \)
Next, we integrate the resulting expression with respect to \( x \):
∫₀² (2x + 3/2) dx
Applying the power rule for integration, we get:
2[x²/2]₀² + 3/2[x]₀² = 2(2 – 0) + 3/2(2 – 0) = 4 + 3 = 7
Final Result
The value of the double integral is \( 7 \).