Problem 58 - Calculus for Biology and Medicine

Question

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{2} x \sqrt{4-x^{2}} d x $$

Step-by-Step Solution

Answer

The definite integral evaluates to \( \frac{8}{3} \).

Step 1: Choose a Substitution

Identify the substitution that simplifies the integral. For the integral \( \int_{0}^{2} x \sqrt{4-x^2} \, dx \), use the substitution \( u = 4 - x^2 \). Thus, \( du = -2x \, dx \).

Step 2: Adjust the Differential

Solve for \( dx \) in terms of \( du \) and \( x \). From \( du = -2x \, dx \), we get \( dx = \frac{du}{-2x} \).

Step 3: Convert Limits of Integration

Substitute the limits of \( x \) to find the corresponding limits for \( u \). When \( x = 0 \), \( u = 4 - 0^2 = 4 \). When \( x = 2 \), \( u = 4 - 2^2 = 0 \). The new limits are from \( u = 4 \) to \( u = 0 \).

Step 4: Substitute into the Integral

Substitute \( u \) and \( dx \) into the integral: \( \int_{4}^{0} x \sqrt{u} \cdot \frac{du}{-2x} \). Simplify to \( -\frac{1}{2} \int_{4}^{0} \sqrt{u} \, du \). Reverse the limits to adjust the sign: \( \frac{1}{2} \int_{0}^{4} u^{1/2} \, du \).

Step 5: Evaluate the Integral

Integrate \( \frac{1}{2} \int_{0}^{4} u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is \( \frac{2}{3}u^{3/2} \). Thus, \( \frac{1}{2} \left[ \frac{2}{3}u^{3/2} \right]_{0}^{4} \).

Step 6: Evaluate at Bounds

Apply the Fundamental Theorem of Calculus by evaluating at the bounds. Plug the limits into the antiderivative: \( \frac{1}{2} \left( \frac{2}{3} (4)^{3/2} - \frac{2}{3} (0)^{3/2} \right) \). Simplify to \( \frac{1}{2} \times \frac{2}{3} \times 8 \).

Step 7: Compute the Final Answer

Calculate the resulting expression: \( \frac{1}{2} \times \frac{2}{3} \times 8 = \frac{8}{3} \). Thus, the value of the definite integral is \( \frac{8}{3} \).

Key Concepts

Substitution Method

The substitution method is a useful technique for evaluating integrals, making integration easier. It involves replacing a part of the integral with a new variable, which simplifies the problem. In the exercise provided, we use the substitution \( u = 4 - x^2 \). Here’s why this is such a clever choice:

• The expression under the square root, \( 4-x^2 \), becomes a simple variable \( u \), making the integral easier to work with.
• The derivative of the substitution, \( du = -2x \, dx \), gives an expression for \( dx \), which is \( dx = \frac{du}{-2x} \).

This substitution not only simplifies the mathematical operations but also transforms the limits of integration, from \( x \) values to \( u \) values, creating a pathway to solve the integral more efficiently. The next time you see a complex integral, try the substitution method to potentially make it simpler!

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, showing that they are essentially inverse operations. It has two main parts, but here we focus on the part that assists in evaluating definite integrals. This theorem states that if \( F(x) \) is an antiderivative of \( f(x) \), then \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

In this exercise, once we simplified the integral to \( \frac{1}{2} \int_{0}^{4} u^{1/2} \, du \), we could easily find the antiderivative. The antiderivative of \( u^{1/2} \) is \( \frac{2}{3} u^{3/2} \). With the theorem, we apply the antiderivative to the new limits (from 0 to 4).

• Evaluate the antiderivative at the upper limit: \( \frac{2}{3} \, (4)^{3/2} \).
• Evaluate the antiderivative at the lower limit: \( \frac{2}{3} \, (0)^{3/2} \).
• Subtract these two results and multiply by any constants from the simplification process, here \( \frac{1}{2} \).

This is how the fundamental theorem provides a straightforward way to find the exact value of a definite integral.

Integration by Substitution

Integration by substitution is essentially the reverse action of the chain rule from calculus differentiation, and it's a powerful tool for solving integrals. Here's how it plays out in this context.

When we first identified \( u = 4 - x^2 \), the aim was to change the variable of integration so that the integral becomes one we can easily solve. By substituting \( u \) into the integral, it changes the complex original integral into an easier form:

• The original integral: \( \int_{0}^{2} x \sqrt{4-x^2} \, dx \)
• After substitution: \(-\frac{1}{2} \int_{4}^{0} \sqrt{u} \, du \)
• With adjusted limits: \( \frac{1}{2} \int_{0}^{4} u^{1/2} \, du \)

The beauty of integration by substitution is in how it can convert an intimidating integral into one that’s straightforward to manage. By making thoughtful substitutions and appropriate changes to the limits and differentials, you unlock a simpler path to the solution, ensuring an efficient and accurate computation of the integral.