Which of the following definite integrals could be used to calculate the total area bounded by the graph of y = 1 – x2 and the x-axis?

the integral from 0 to 1 of the quantity 1 minus x squared, dx plus the integral from 1 to 2 of the quantity 1 minus x square, dx
the integral from 0 to 1 of the quantity 1 minus x squared, dx minus the integral from 1 to 2 of the quantity 1 minus x square, dx
the integral from 0 to 2 of the quantity 1 minus x squared, dx
2 times the integral from 0 to 1 of the quantity 1 minus x squared, dx

1 answer

Because of the nice symmetry
(and the x-intercepts -1 and +1)

area = 2 ∫(1 - x^2) dx from 0 to 1

which matches the last choice