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
What is the answer to the problem
The definite integral that could be used to calculate the total area bounded by the graph of y = 1 - x^2 and the x-axis is:
The integral from 0 to 2 of the quantity 1 minus x squared, dx (option C)
To calculate the total area bounded by the graph of y = 1 - x^2 and the x-axis, we need to find the area between the curve and the x-axis in the given interval.
The correct integral to calculate the area between the curve and the x-axis from 0 to 1 is ∫(0 to 1) (1 - x^2) dx. This integral represents the area under the curve y = 1 - x^2 from x = 0 to x = 1.
The correct integral to calculate the area between the curve and the x-axis from 1 to 2 is ∫(1 to 2) (1 - x^2) dx. This integral represents the area under the curve y = 1 - x^2 from x = 1 to x = 2.
Since we are looking for the total area bounded by the graph and the x-axis, we need to sum up these two integrals:
∫(0 to 1) (1 - x^2) dx + ∫(1 to 2) (1 - x^2) dx
Therefore, the correct option is "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 correctly accounts for both intervals and calculates the total area.
you know the limits of integration are [-1,1]
None of the choices uses that, but you also know the region is symmetric about the y-axis.
That leaves only one choice.
what do you have against actual math notation? 1-x^2. Use symbols instead of words where you can. Makes things a lot less noisy.