Which integral gives the area of the region in the first quadrant bounded by the graphs of y2 = x − 1, x = 0, y = 0, and y = 2?

the integral from 0 to 1 of 2, dx plus the integral from 1 to 5 of the quantity 2 minus square root of x minus 1, dx

the integral from 0 to 5 of 2 minus the square root of x minus 1, dx

the integral from 0 to 1 of 2, dx plus the integral from 1 to 5 of the quantity 3 minus x, dx

the integral from 0 to 1 of 2, dx minus the integral from 1 to 5 of the quantity 2 minus square root of x minus 1, dx

Interesting that only integrals along the -axis are suggested when integrating along the -axis would be much simpler... Anyway, you have to split the interval of integration into two. The "height" of the region is not uniform over the entire interval.

When , we have . When , we have . Then the area we want is given by

which seems to agree with the last option.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

answer: what is the conflict?

step-by-step explanation:

*what has happened to new york city?

*what is in the time capsule?

* what did the kidz see that gave them hope?

*what will they do with what they find?

* how will they save the city?

* what challenges will they face along the way?

*what big choice will they make at the climax of the story?

* what lesson do they learn at the end of the story?