Find the area between the curves y = x^2 and y = 4 from x = -2 to x = 2.
Practice Questions
Q1
Find the area between the curves y = x^2 and y = 4 from x = -2 to x = 2.
8/3
16/3
8
4
Questions & Step-by-Step Solutions
Find the area between the curves y = x^2 and y = 4 from x = -2 to x = 2.
Step 1: Identify the curves. We have y = x^2 (a parabola) and y = 4 (a horizontal line).
Step 2: Determine the points of intersection. Set x^2 = 4 to find where the curves meet. This gives x = -2 and x = 2.
Step 3: Set up the integral to find the area between the curves. The area is given by the integral of the top curve minus the bottom curve from x = -2 to x = 2. Here, the top curve is y = 4 and the bottom curve is y = x^2.
Step 4: Write the integral: Area = ∫(from -2 to 2) (4 - x^2) dx.
Step 5: Calculate the integral. First, find the antiderivative of (4 - x^2), which is 4x - (x^3)/3.
Step 6: Evaluate the antiderivative from -2 to 2. Plug in 2: (4*2 - (2^3)/3) and plug in -2: (4*(-2) - ((-2)^3)/3).
Step 7: Calculate the values: For x = 2, you get 8 - 8/3 = 24/3 - 8/3 = 16/3. For x = -2, you get -8 + 8/3 = -24/3 + 8/3 = -16/3.
Step 8: Subtract the two results: (16/3) - (-16/3) = 16/3 + 16/3 = 32/3.
Step 9: The area between the curves from x = -2 to x = 2 is 32/3.
Area Between Curves – Calculating the area between two curves involves integrating the difference of the functions over a specified interval.
Definite Integrals – Understanding how to evaluate definite integrals and apply the Fundamental Theorem of Calculus.
Identifying Functions – Recognizing which function is above the other in the given interval to set up the correct integral.
