Evaluate the integral by interpreting it in terms of areas. integral -4 to 0 (3+Radical(16-x^2))dx

Question

Evaluate the integral by interpreting it in terms of areas. integral -4 to 0 (3+Radical(16-x^2))dx

Answer 1

Interesting! I don't know how to integrate the expression as given, so I made a plot of it: this function is part of a downwards parabola with vertex at (0,7) and left end at (-4,3). Setting y = a x² + 7 yields a = -1/4. Then ∫_-4⁰ (- x²/4 + 7) dx = [- x³/12 + 7x]_-4⁰ = 0 - (16/3 - 28) = 68/3 ≅ 22.7 for the area of this curve above the negative x axis.