Calculus Question

Question

. Using the Fundamental Theorem of Calculus find the area of the region bounded by the xx-axis and the graph of f(x)=−x2+0x+4 

Answer: 

Answer 1

The graph of f(x) = -x² + 4 has roots at x = -2 and x = +2.

Between those two points the value of f(x) is positive.

So the area to be determined is the area below the curve of f(x) between the two roots.

The indefinite ∫f(x)dx = -x³/3 + 4x + C. Call F(x) = -x³/3 + 4x.

The area to be determined is the definite ∫f(x)dx evaluated between x = -2 and x = +2,

that is,

F(2) - F(-2) = (-2³/3 + 4×2) - (-(-2)³/3 + 4×(-2)) = 32/3