Use the limit process to find the area of the region between the graph of the function f(x)= x^3 and the x-axis on the interval [0,2]
To find the area of the region between the graph of the function f(x) = x^3 and the x-axis on the interval [0, 2], we can use the limit process. Here are the steps to do so:
1. Divide the interval [0, 2] into smaller subintervals. Let's call the width of each subinterval "Δx".
2. Choose any point within each subinterval and evaluate the function f(x) = x^3 at that point. Let's denote this point as "xi" for the i-th subinterval.
3. Multiply the value of f(xi) by the width of the subinterval (Δx). This will give you the area of a rectangle that represents the approximate area between the graph of f(x) and the x-axis within that subinterval. Let's denote this area as "Ai".
4. Repeat steps 2 and 3 for all the subintervals to find the approximate areas corresponding to each subinterval.
5. Add up all the approximate areas (Ai) to get the total approximate area of the region between the graph of f(x) and the x-axis on the interval [0, 2].
6. To find the exact area, take the limit as the number of subintervals approaches infinity (Δx approaches 0) of the sum of all the approximate areas (Ai).
Mathematically, the exact area can be obtained as follows:
Area = ∫(0 to 2) f(x) dx
Using the function f(x) = x^3, we can integrate it over the interval [0, 2] to find the exact area. The antiderivative of x^3 is (1/4)x^4, so the final calculation would be:
Area = ∫(0 to 2) x^3 dx
= [ (1/4)x^4 ] (0 to 2)
= (1/4)(2^4) - (1/4)(0^4)
= (1/4)(16)
= 4 square units
Therefore, the area of the region between the graph of f(x) = x^3 and the x-axis on the interval [0, 2] is 4 square units.