Question

Find the area betweren f(x)=x^(2)+2 and g(x)=1-x between x=0 and x=1

Answer 1

To find the area between two curves, calculate the integral of each curve and subtract the smaller integral from the larger integral. In this case, the area between f(x) = x^2 + 2 and g(x) = 1 - x between x=0 and x=1 is (7/6) square units.

Step-by-step explanation:

To find the area between two curves, we need to find the area of the region bounded by the curves. In this case, the two curves are f(x) = x^2 + 2 and g(x) = 1 - x. The area between the curves can be found by subtracting the area under g(x) from the area under f(x) between the given limits of x=0 and x=1. Here are the steps to calculate the area:

1. Calculate the integral of f(x) from x=0 to x=1: ∫(x^2 + 2) dx
2. Calculate the integral of g(x) from x=0 to x=1: ∫(1 - x) dx
3. Subtract the area under g(x) from the area under f(x): ∫(x^2 + 2) dx - ∫(1 - x) dx
4. Simplify the expression and calculate the definite integral: ∫(x^2 + 2) dx - ∫(1 - x) dx = (1/3)x^3 + 2x - (x - (1/2)x^2)
5. Evaluate the expression using the given limits of x=0 and x=1: ((1/3)(1)^3 + 2(1) - (1 - (1/2)(1)^2)) - ((1/3)(0)^3 + 2(0) - (0 - (1/2)(0)^2))
6. Simplify the expression: (1/3 + 2 - (1 - 1/2)) - (0)
7. Calculate the final result: (7/6) square units