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
does the interval extend to x=2?
To calculate the total area bounded by the graph of y = 1 - x^2 and the x-axis, we need to find the definite integral of the function between the appropriate limits.
Option 1:
The integral from 0 to 1 of the quantity 1 minus x squared, dx represents the area between the graph and the x-axis from x = 0 to x = 1.
Then, the integral from 1 to 2 of the quantity 1 minus x squared, dx represents the area between the graph and the x-axis from x = 1 to x = 2.
So, option 1 is a valid method to find the total area bounded by the graph.
Option 2:
Similar to option 1, the integral from 0 to 1 of the quantity 1 minus x squared, dx represents the area from x = 0 to x = 1.
However, subtracting the integral from 1 to 2 of the quantity 1 minus x squared, dx means subtracting the area between the graph and the x-axis from x = 1 to x = 2.
Therefore, option 2 is not a valid method to find the total area.
Option 3:
The integral from 0 to 2 of the quantity 1 minus x squared, dx represents the area between the graph and the x-axis from x = 0 to x = 2.
Thus, option 3 is also a valid method to find the total area.
Option 4:
Multiplying 2 with the integral from 0 to 1 of the quantity 1 minus x squared, dx would give us twice the area from x = 0 to x = 1.
But this does not take into account the area between x = 1 and x = 2.
Hence, option 4 is not a valid method to find the total area.
In conclusion, options 1 and 3 are the valid definite integrals that could be used to calculate the total area bounded by the graph of y = 1 - x^2 and the x-axis.