Question

For the region formed by the functions f(x)=x−1 and g(x)=−2 on the interval [−3, 1], use definite integrals to find the area of the region.

Answer 1

The area of the region formed by f(x) = x - 1 and g(x) = -2 on the interval [-3, 1] is found by integrating the function (x + 1) from -3 to 1, which represents the area under the curve between these two functions.

Step-by-step explanation:

The student is asking how to find the area of the region bounded by the functions f(x) = x - 1 and g(x) = -2 on the interval [−3, 1]. To obtain this area using definite integrals, we can integrate the difference between f(x) and g(x) over the specified interval.

Step-by-Step Solution:

1. Identify the top and bottom functions. In this case, f(x) is above g(x) on the interval.
2. Write the definite integral for the area: ∫₁ₒ₃ (f(x) - g(x)) dx, which simplifies to ∫₁ₒ₃ (x - 1 - (-2)) dx.
3. Evaluate the integral from -3 to 1: ∫₁ₒ₃ (x + 1) dx.
4. The result of the integration will give us the area of the region. Carry out the integration and compute the definite integral to obtain the final answer.

By calculating the area under the curve between f(x) and g(x) using definite integrals, the student will understand the geometric interpretation of the integral as the accumulated sum of infinitesimal areas along the x-axis.