Question

Identify each expression and value that represents the area under the curve (y = x² + 4) on the interval ([-3, 2]).

a) (int₋₃² (x² + 4) , dx)

b) (int₋₃² x² , dx)

c) (int₋₃² 4 , dx)

d) (int₋₃² 0 , dx)

Answer 1

The question involves determining the area under the curve y = x² + 4 using definite integration over the interval [-3, 2]. The area is found by evaluating the integral of the function from -3 to 2 and applying the Fundamental Theorem of Calculus.

Step-by-step explanation:

The student has asked to identify expressions and values that represent the area under the curve for the function y = x² + 4 on the interval [-3, 2]. To determine the area under the curve, we use definite integration over the given interval.

• a) ∫⁻₃² (x² + 4) dx represents the total area under the curve y = x² + 4 from x = -3 to x = 2.
• b) ∫⁻₃² x² dx represents the area under the curve for the portion of the function y = x² from x = -3 to x = 2.
• c) ∫⁻₃² 4 dx represents the area under the horizontal line y = 4 across the same interval.
• d) ∫⁻₃² 0 dx would simply yield a value of 0, as there is no area under the curve y = 0.

To calculate these integrals step by step, you would:

1. Set up the definite integral with limits of integration from -3 to 2.
2. Integrate the function within the interval.
3. Apply the Fundamental Theorem of Calculus by plugging in the upper limit into the integral, subtracting the value obtained by plugging in the lower limit.

For instance, the integral in (a) would be evaluated as:

1. Find the integral of x² + 4 which is ⅓x³ + 4x.
2. Evaluate ⅓x³ + 4x at x = 2.
3. Evaluate ⅓x³ + 4x at x = -3 and subtract from the step 2 result.