find the area of the region bounded by the graphs of y=-x^2+5x+2 and y=2
The graphs intersect at (0,2) and (5,2). So, the area is just the region below the parabola and above the line:
a = ∫[0,5] y-2 dx
= ∫[0,5] (-x^2+5x+2-2) dx = 125/6
THANK YOU!!!!!!!
Why did the area go to the party last night? Because it couldn't resist the lure of bounded curves and calculating the integral! To find the area of the region bounded by those two graphs, we need to set them equal to each other and solve for the x-values where they intersect. So, let's set -x^2 + 5x + 2 = 2 and solve for x.
To find the area of the region bounded by the graphs of y = -x^2 + 5x + 2 and y = 2, we need to find the x-values where these two curves intersect and calculate the definite integral of the difference between the two functions over that interval. Here are the step-by-step instructions:
Step 1: Set the two functions equal to each other to find the x-values where they intersect:
-x^2 + 5x + 2 = 2
Step 2: Simplify the equation:
-x^2 + 5x = 0
Step 3: Factor out x:
x(-x + 5) = 0
Step 4: Solve for x by setting each factor equal to zero:
x = 0 and -x + 5 = 0
x = 0 and x = 5
Step 5: Determine the interval over which we will calculate the area:
Since the graph of y = -x^2 + 5x + 2 is below the graph of y = 2, we need to integrate from x = 0 to x = 5.
Step 6: Set up the integral to calculate the area:
Area = ∫[0, 5] (2 - (-x^2 + 5x + 2)) dx
Step 7: Simplify the integral:
Area = ∫[0, 5] (x^2 - 5x) dx
Step 8: Integrate the function:
Area = [ (1/3)x^3 - (5/2)x^2 ] evaluated from 0 to 5
Step 9: Calculate the definite integral:
Area = [(1/3)(5^3) - (5/2)(5^2)] - [(1/3)(0^3) - (5/2)(0^2)]
Step 10: Simplify the expression:
Area = [125/3 - 125/2] - [0 - 0]
Step 11: Calculate the final result:
Area = (125/3) - (125/2)
The area of the region bounded by the given curves is (125/3) - (125/2) square units.
To find the area of the region bounded by the graphs of the given equations, we need to determine the points where the graphs intersect and set up an integral to calculate the area.
Step 1: Find the points of intersection:
Set the two equations equal to each other and solve for x:
-x^2 + 5x + 2 = 2
Simplifying the equation, we get:
-x^2 + 5x = 0
x(-x + 5) = 0
From here, we have two possible solutions:
x = 0 (First intersection point)
or
-x + 5 = 0
x = 5 (Second intersection point)
Step 2: Set up the integral:
To find the area, we integrate the difference between the two curves with respect to x, between the two intersection points found.
We integrate (y1 - y2) with respect to x, where y1 represents the equation y = -x^2 + 5x + 2 and y2 represents the equation y = 2.
∫[0 to 5] (-x^2 + 5x + 2 - 2) dx
Step 3: Evaluate the integral:
Integrating the expression, we get:
Area = ∫[0 to 5] (-x^2 + 5x) dx
= [-x^3/3 + (5x^2)/2] evaluated from 0 to 5
= [-125/3 + 125/2] - [0]
= [125/2 - 125/3]
= 375/6 - 250/6
= 125/6
Therefore, the area of the region bounded by the graphs of y = -x^2 + 5x + 2 and y = 2 is 125/6 square units.