1. Find the are between the curves y=e^x and y=4-x^2 graphically.
a.) set up the integral
b.) include bounds rounded to three decimal places
c.) use integral function on calculator
2. Find the area of the "triangular" region bounded by y=4-x on the left, y=sqrt(x-2) on the right and y=2 on the top. Set up the integral and use a calculator.
The curves intersect at x = -1.965 and x=1.058
so the area between the curves is
∫[-1.965,1.058] ((4-x^2)-e^x) dx
To do this using vertical strips of width dx involves changing boundaries at x=3. So instead, let's use horizontal strips of width dy, so we get
∫[1,2] ((y^2+2)-(4-y)) dy
To find the area between curves graphically, follow these steps:
1. Graph both curves on the same set of axes.
- For the first problem, graph the curves y = e^x and y = 4 - x^2.
- For the second problem, graph the curves y = 4 - x and y = sqrt(x - 2), as well as the line y = 2.
2. Identify the points where the two curves intersect.
- For the first problem, find the intersections of y = e^x and y = 4 - x^2.
- For the second problem, find the intersections of y = 4 - x and y = sqrt(x - 2).
3. Determine the region enclosed between the curves.
- For the first problem, this is the region between the curves y = e^x and y = 4 - x^2.
- For the second problem, this is the triangular region bounded by y = 4 - x, y = sqrt(x - 2), and y = 2.
4. Set up the integral to find the area between the curves.
- For the first problem, the integral represents the area between the curves y = e^x and y = 4 - x^2. The integral can be set up as ∫(4 - x^2 - e^x) dx.
- For the second problem, the integral represents the area of the triangular region. The integral can be set up as ∫(4 - x - sqrt(x - 2) - 2) dx.
5. Determine the bounds for the integral.
- To find the bounds, set the rightmost and leftmost x-values where the curves intersect as the lower and upper limits of integration.
- Round the bounds to three decimal places as instructed.
6. Calculate the area using an integral function on a calculator.
- Use a calculator with an integral function to evaluate the integral and find the area between the curves.
- Enter the integral expression (with proper bounds) into the calculator and evaluate the integral to get the answer.
Remember to follow these steps for both problems to find the requested areas.