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
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 the absolute value of the function over the desired interval.
The definite integral represents the area under the curve between two points on the x-axis. So, we can split the given function into two parts, from 0 to 1 and from 1 to 2, and calculate the integral for each part separately.
Looking at the options, we can rule out the last option "2 times the integral from 0 to 1 of the quantity 1 minus x squared, dx" since it multiplies the first part of the integral by 2, which doubles the area.
Now let's consider the remaining options:
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 squared, dx
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 squared, dx
3. The integral from 0 to 2 of the quantity 1 minus x squared, dx
To calculate the total area, we should take the sum of the areas in the given intervals. So, the correct option is the first one:
"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 squared, dx".
This option correctly splits the integral into two separate integrals for the intervals from 0 to 1 and from 1 to 2, and sums up the areas of those intervals to calculate the total area bounded by the graph.