Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 32. y 2 21 s4x 2 1 x 1 2d dx 33. y 1 0 sx 3 2 3x 2 d dx 34. y 2 0 s2x 2 x 3 d dx

To evaluate the given integrals using the form of the definition of the integral (as in Theorem 4, which typically involves using the Fundamental Theorem of Calculus), we need to compute the definite integrals over the specified intervals.

Let's rewrite and solve each integral step by step:

### Problem 32

Evaluate the integral:
\[ \int_{2}^{21} (4x^2 + 1) \, dx \]

1. **Integrate the function \(4x^2 + 1\)**:
   \[
   \int (4x^2 + 1) \, dx = \int 4x^2 \, dx + \int 1 \, dx = \frac{4x^3}{3} + x + C
   \]

2. **Evaluate the definite integral from \(2\) to \(21\)**:
   \[
   \left[ \frac{4x^3}{3} + x \right]_{2}^{21} = \left( \frac{4 \cdot 21^3}{3} + 21 \right) - \left( \frac{4 \cdot 2^3}{3} + 2 \right)
   \]

3. **Calculate the values**:
   \[
   \frac{4 \cdot 21^3}{3} + 21 = \frac{4 \cdot 9261}{3} + 21 = \frac{37044}{3} + 21 = 12348 + 21 = 12369
   \]
   \[
   \frac{4 \cdot 2^3}{3} + 2 = \frac{4 \cdot 8}{3} + 2 = \frac{32}{3} + 2 = \frac{32}{3} + \frac{6}{3} = \frac{38}{3} = 12.67
   \]

4. **Subtract the lower limit from the upper limit**:
   \[
   12369 - 12.67 = 12356.33
   \]

So, the value of the integral is:
\[ \int_{2}^{21} (4x^2 + 1) \, dx = 12356.33 \]

### Problem 33

Evaluate the integral:
\[ \int_{1}^{0} (x^3 - 3x^2) \, dx \]

1. **Integrate the function \(x^3 - 3x^2\)**:
   \[
   \int (x^3 - 3x^2) \, dx = \int x^3 \, dx - \int 3x^2 \, dx = \frac{x^4}{4} - x^3 + C
   \]

2. **Evaluate the definite integral from \(1\) to \(0\)**:
   \[
   \

Learn how to evaluate definite integrals using the form of the definition of the integral as given in Theorem 4. Detailed examples and solutions provided for integrals such as ∫(4x^2 + 1) dx from 2 to 21, ∫(x^3 - 3x^2) dx from 1 to 0, and more. Suitable for undergraduate students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation