Question

Evaluate 6 −6 (x + 6) 36 − x2 dx by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.

Answer 1

The integral 6 - 6(x + 6) + 36 - x^2 dx can be split into two separate integrals for evaluation: a linear function and a quadratic function representing an area under a curve. The areas are interpreted by taking the definite integral with specified limits.

Step-by-step explanation:

To evaluate the integral 6 − 6 (x + 6) + 36 − x^2 dx, let's first correct the expression assuming it is ∫ (6 - 6(x + 6) + 36 - x^2) dx and then write it as a sum of two integrals:

1. ∫ (6 - 6(x + 6) + 36) dx, which simplifies to ∫ (36 - 6x) dx, a standard linear function that can be directly integrated.

2. ∫ (-x^2) dx, which represents the area under the curve y = -x^2 from the given integral limits, assuming that the proper limits are provided.

Once the limits are set, for instance, from x = a to x = b, the area corresponding to the second integral can be interpreted as the area between the curve y = -x^2 and the x-axis within those limits. Remember that areas below the x-axis are considered negative when calculating the definite integral.

The significance of each component within the integral is to identify the geometric or physical interpretations, such as work done, force, or probability, depending on the context provided by the question. Here we are focusing on the mathematical calculation and area interpretation.

Answer 2

I think this is the answer for your answer -216x - 1290 - dx^3. I hope this is the right answer for you. Let me on if it's the wrong answer for me.