AP Calculus AB Exam Question:

Determine the area of the region bounded by the curves 𝑦 = 𝑥^2 and 𝑦 = 2𝑥.

Step-by-step Solution:

To find the area between two curves, we need to locate the points where they intersect first.

Setting the two equations equal to each other, we can find the x-values of the intersection points:

𝑥^2 = 2𝑥

Rearranging the equation:

𝑥^2 - 2𝑥 = 0

Factoring out an 𝑥:

𝑥(𝑥 - 2) = 0

Therefore, the two intersection points are 𝑥 = 0 and 𝑥 = 2.

Next, we need to determine the limits of integration. Since 𝑦 = 𝑥^2 is below 𝑦 = 2𝑥 in the interval [0, 2], the limits of integration will be 0 and 2.

Now, we want to find the area between the curves, so we'll evaluate the definite integral:

∫[0,2] (2𝑥 - 𝑥^2) 𝑑𝑥

Expanding the integrand, we have:

∫[0,2] (2𝑥)𝑑𝑥 - ∫[0,2] (𝑥^2)𝑑𝑥

Integrating each term separately:

∫[0,2] (2𝑥)𝑑𝑥 = 𝑥^2 |[0,2] = (2^2) - (0^2) = 4 - 0 = 4

∫[0,2] (𝑥^2)𝑑𝑥 = (1/3)𝑥^3 |[0,2] = (1/3)(2^3) - (1/3)(0^3) = (1/3)(8) - (0) = 8/3

Substituting the integration results back into the original integral, we have:

4 - 8/3 = 12/3 - 8/3 = 4/3

Therefore, the area between the curves 𝑦 = 𝑥^2 and 𝑦 = 2𝑥 in the interval [0, 2] is 4/3 square units.