Pleaseee help on these two questions!
1. Use geometry to evaluate integral(with upper bound of 10 and lower bound of 0) of f(x) dx for f(x)= {5, x <=5 and 10-x, x > 5
A. 12.5
B. 25
C. 37.5
D. Cannot be found
2. Use your calculator to evaluate the limit from x equals 0 to 2 of sin(x^2) dx. Give your answer to the nearest integer.
______(fill in the blank)
Thank you in advance for the help. I greatly appreciate it, Thank you!
Oobleck, can you please help me out on one more question. I'm gonna post it right now. Thanks!
The response above is mine. Just figured I'd let you know so that you don't get confused. Thank you!
Oobleck, I got 25 for answer one. I got it marked wrong. I guess I did something wrong along the way. Can you please explain to me the process. I don't know where I messed up while working the problem.
The square has an area of 25
The triangle to the right has an area of 12.5
Clearly you did not draw the graph, nor read my hint.
Sure! I'd be happy to help you with these questions.
1. To evaluate the integral of f(x) with the given bounds, we need to divide the area under the curve into two parts: one where x is less than or equal to 5, and another where x is greater than 5. Let's calculate the area for each part separately.
First, let's find the area for x ≤ 5. Since f(x) is a constant 5 in this range, the area is simply the base multiplied by the height of the rectangle. So, the area for x ≤ 5 is (5 * 5) = 25.
Next, let's find the area for x > 5. In this range, f(x) is given by (10 - x), which forms a linear decreasing function. The area under this curve can be visualized as a triangle with height 5 and base (10 - 5) = 5. Therefore, the area for x > 5 is (0.5 * 5 * 5) = 12.5.
Now, to find the total area, we add the areas for both parts: 25 + 12.5 = 37.5.
Therefore, the answer to the integral is option (C) 37.5.
2. To evaluate the given limit, we need to use a calculator that supports numerical integration. Here's how you can do it:
- Open your calculator and switch to the integration mode.
- Enter the function sin(x^2).
- Set the lower bound as 0 and the upper bound as 2.
- Calculate the integral.
Since the question asks for the answer to the nearest integer, round the result to the nearest whole number.
Therefore, you need to use a calculator or a tool that supports numerical integration to find the integral value. Unfortunately, as an AI text-based bot, I don't have access to a calculator. So, I'm unable to provide you with the exact value. However, you can follow the steps I mentioned above using your calculator to find the value and round it to the nearest integer.
I hope this helps you with your questions! Let me know if there's anything else I can assist you with.
#1 Really? Draw the graph. You have a square and a triangle. What do you get?
#2 So, use your calculator
You can also verify your result here:
https://www.wolframalpha.com/input/?i=integral%5B0..2%5D+sin(x%5E2)+dx