Calculas Section 1.7 Question 15

Introduction

Calculas Section 1.7 Question 15 typically involves solving a definite integral or evaluating a limit. This question tests your understanding of integral calculus and your ability to apply integration techniques to solve real-world problems.

The integral in question is often of the form ∫[a to b] f(x) dx, where f(x) might be a polynomial, trigonometric function, or exponential function. The solution requires finding the antiderivative of f(x) and evaluating it at the bounds a and b.

Formula

The fundamental theorem of calculus states that if f(x) is continuous on the closed interval [a, b], then the definite integral of f(x) from a to b can be evaluated as:

Where F(x) is the antiderivative of f(x), meaning F'(x) = f(x).

For polynomial functions, the antiderivative is found by increasing each exponent by one and dividing by the new exponent. For trigonometric functions, standard antiderivative formulas apply.

Worked Example

Let's solve the integral ∫[0 to 2] (3x² + 2x) dx step by step.

1. Find the antiderivative F(x) of f(x) = 3x² + 2x:
   F(x) = ∫(3x² + 2x) dx = x³ + x² + C
2. Evaluate F(x) at the bounds:
   F(2) = (2)³ + (2)² = 8 + 4 = 12
   F(0) = (0)³ + (0)² = 0 + 0 = 0
3. Subtract the lower bound from the upper bound:
   ∫[0 to 2] (3x² + 2x) dx = F(2) - F(0) = 12 - 0 = 12

The definite integral evaluates to 12.

Interpreting Results

The result of a definite integral represents the net accumulation of the function f(x) over the interval [a, b]. For example, if f(x) represents a rate of change (like velocity), the integral gives the net change (like displacement).

When solving Calculas Section 1.7 Question 15, consider:

• The physical meaning of the integral in the context of the problem
• Whether the result makes sense in the real-world scenario
• How the integral compares to other similar problems you've solved

If your result seems counterintuitive, double-check your antiderivative and evaluation steps.