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
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
To calculate the total area bounded by the graph of y = 1 – x^2 and the x-axis, we need to find the integral of 1 – x^2 over the appropriate limits.
The graph of y = 1 – x^2 is a parabola that opens downwards, reaching the x-axis at x = -1 and x = 1. Thus, the area is bounded by the x-axis from x = -1 to x = 1.
Option 1: 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.
This option includes two separate integrals, one from 0 to 1 and the other from 1 to 2. However, we only need the integral from -1 to 1 to calculate the total area. So, this option is not the correct choice.
Option 2: 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.
Similar to the previous option, this choice also includes unnecessary integrals. Therefore, this option is incorrect.
Option 3: The integral from 0 to 2 of the quantity 1 minus x squared, dx.
This option covers the entire interval from -1 to 1, which is the region of interest. Thus, this option is a valid choice to calculate the total area bounded by the graph and the x-axis.
Option 4: 2 times the integral from 0 to 1 of the quantity 1 minus x squared, dx.
Multiplying the integral by 2 does not change the value of the area calculation. Thus, this option is equivalent to the correct choice mentioned above.
Therefore, the correct answer is option 3: the integral from 0 to 2 of the quantity 1 minus x squared, dx.