use the fundamental theorem to find the area under f(x)=x^2 between x=1 and x=4.
how do i go about doing this? i know how to use my calculator, but it says to use the theorem so i'm not completely sure.
The indefinite integral of x^2 is x^3/3. Calculate the vlalue of that function at x = 4 and subtract the value at x= 1. The difference is called the definite integral and will be equal to the area under the y = x^2 curve (measured above the x axis).
To find the area under the curve of f(x) = x^2 between x = 1 and x = 4 using the so-called Fundamental Theorem of Calculus, you can follow these steps:
1. Start by finding the antiderivative (also known as the indefinite integral) of f(x). In this case, the antiderivative of f(x) = x^2 is F(x) = (1/3)x^3.
2. Next, evaluate the antiderivative at the upper and lower bounds of integration. Plug in x = 4 and x = 1 into the antiderivative function to find the values of the antiderivative at these points.
F(4) = (1/3)(4)^3 = 64/3
F(1) = (1/3)(1)^3 = 1/3
3. Subtract the value at the lower bound from the value at the upper bound to find the definite integral. The definite integral represents the area under the curve between the given bounds.
Area = F(4) - F(1) = 64/3 - 1/3 = 63/3 = 21
So, the area under the curve f(x) = x^2 between x = 1 and x = 4 is 21 square units.