Integral from 0 to 3.5 left parenthesis 88200 pi left parenthesis 7 minus y right parenthesis right parenthesis dy

To evaluate the integral 

\[
\int_{0}^{3.5} 88200 \pi (7 - y) \, dy,
\]

let's follow these steps:

1. **Factor out constants:** Since \(88200 \pi\) is a constant, it can be factored out of the integral:

   \[
   88200 \pi \int_{0}^{3.5} (7 - y) \, dy.
   \]

2. **Integrate the function:** Now, integrate the function \(7 - y\) with respect to \(y\):

   \[
   \int (7 - y) \, dy = 7y - \frac{y^2}{2}.
   \]

3. **Apply the limits of integration:** Evaluate the antiderivative from \(0\) to \(3.5\):

   \[
   \left[ 7y - \frac{y^2}{2} \right]_{0}^{3.5}.
   \]

   Substituting the upper limit \(3.5\):

   \[
   7(3.5) - \frac{(3.5)^2}{2} = 24.5 - \frac{12.25}{2} = 24.5 - 6.125 = 18.375.
   \]

   Substituting the lower limit \(0\):

   \[
   7(0) - \frac{(0)^2}{2} = 0.
   \]

   So, the result of the definite integral is:

   \[
   18.375 - 0 = 18.375.
   \]

4. **Multiply by the constant:** Finally, multiply the result by \(88200 \pi\):

   \[
   88200 \pi \times 18.375.
   \]

Let's compute the final value:

\[
88200 \times 18.375 \times \pi.
\]The value of the integral 

\[
\int_{0}^{3.5} 88200 \pi (7 - y) \, dy
\]

is approximately \(5,091,500.67\).

Would you like more details, or do you have any questions?

Here are some related questions you might find interesting:

1. How do you evaluate definite integrals with more complex functions?
2. What are the applications of definite integrals in real-world problems?
3. How does the Fundamental Theorem of Calculus relate to evaluating definite integrals?
4. How can you integrate functions involving trigonometric terms?
5. What are some common techniques for integrating products of functions?

**Tip:** Always check if you can simplify the integrand before integrating; factoring out constants can make the process much easier!

Learn how to evaluate the definite integral ∫[0, 3.5] 88200π(7 - y) dy, which involves constant multiplication and applying the Fundamental Theorem of Calculus. This detailed step-by-step solution is suitable for students at the high school or college level 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