Find the area under the curve y = x^2 + 2x from x = 0 to x = 3.

Find the area under the curve y = x^2 + 2x from x = 0 to x = 3.

Step 1: Identify the function you want to find the area under. In this case, the function is y = x^2 + 2x.
Step 2: Set up the integral to find the area under the curve from x = 0 to x = 3. This is written as ∫(from 0 to 3) (x^2 + 2x) dx.
Step 3: Calculate the integral of the function. The integral of x^2 is (x^3)/3 and the integral of 2x is x^2. So, the integral of (x^2 + 2x) is (x^3)/3 + x^2.
Step 4: Evaluate the integral from the lower limit (0) to the upper limit (3). This means you will calculate [(3^3)/3 + (3^2)] - [(0^3)/3 + (0^2)].
Step 5: Calculate (3^3)/3, which is 27/3 = 9, and (3^2), which is 9. So, you have 9 + 9 = 18.
Step 6: Since the lower limit evaluates to 0, you subtract 0 from 18, which gives you the final area under the curve as 18.

Definite Integral – The process of calculating the area under a curve by evaluating the integral of a function over a specified interval.
Polynomial Functions – Understanding the properties and behavior of polynomial functions, specifically quadratic functions in this case.
Fundamental Theorem of Calculus – The theorem that connects differentiation and integration, allowing for the evaluation of definite integrals using antiderivatives.