B.Integral from 0 to 1 left parenthesis 9800 left parenthesis 6 minus y right parenthesis right parenthesis dy
Integral from 0 to 1 left parenthesis 9800 left parenthesis 6 minus y right parenthesis right parenthesis dy

To solve the integral

\[
\int_{0}^{1} 9800 (6 - y) \, dy,
\]

we can first factor out the constant \(9800\) from the integral:

\[
9800 \int_{0}^{1} (6 - y) \, dy.
\]

Now, let's compute the integral of \(6 - y\) with respect to \(y\).

### Step-by-Step Solution

1. **Find the indefinite integral:**

\[
\int (6 - y) \, dy = 6y - \frac{y^2}{2} + C.
\]

2. **Evaluate the definite integral from 0 to 1:**

\[
9800 \left[ \left(6y - \frac{y^2}{2}\right) \Bigg|_{0}^{1} \right].
\]

3. **Plug in the limits:**

\[
9800 \left[ \left(6(1) - \frac{1^2}{2}\right) - \left(6(0) - \frac{0^2}{2}\right) \right].
\]

Simplify inside the brackets:

\[
9800 \left[ (6 - 0.5) - 0 \right] = 9800 \times 5.5.
\]

4. **Compute the final value:**

\[
9800 \times 5.5 = 53900.
\]

### Final Answer

\[
\int_{0}^{1} 9800 (6 - y) \, dy = 53900.
\]

Would you like further details, or do you have any questions about this solution?

### Related Questions:
1. What is the meaning of a definite integral in terms of area under a curve?
2. How do you compute the integral of a polynomial function?
3. How would the result change if the integral were from 0 to 2 instead of 0 to 1?
4. What is the effect of the constant multiplier on the integral result?
5. How do you handle integrals with more complex functions, like trigonometric or exponential?

### Tip:
When integrating, always check if there are constants that can be factored out to simplify the calculation.

Integral from 0 to 1 left parenthesis 9800 left parenthesis 6 minus y right parenthesis right parenthesis dy

Learn how to calculate the definite integral of 9800(6 - y) with limits from 0 to 1. This detailed step-by-step solution covers key calculus concepts like factoring constants and applying the fundamental theorem of calculus. Suitable for students in Grades 11-12.

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