Riley! We should be able to figure some integrals geometrically using transformations of functions. {{CMD_7}} That sounds like a cool idea. Maybe, since we know the graph of {{MATH_8}} is a semicircle, we get an ellipse defined on {{MATH_9}} just by stretching the graph of {{MATH_10}} by a factor of {{MATH_11}} horizontally. The equation of this ellipse would be {{MATH_12}} {{CMD_13}} Exactly! So since we know that {{MATH_14}} geometrically{{CMD_15}} {{CMD_16}} And we know that the area under {{MATH_17}} from {{MATH_18}} is twice the under {{MATH_19}}{{CMD_20}} {{CMD_21}} We must have {{MATH_22}} {{CMD_23}} Jinx! {{CMD_24}} It is kind of like we just stretched out our whole coordinate system, and that helped us solve an integral. {{CMD_25}} In this case, everything got stretched out by a constant factor of {{MATH_26}} in the horizontal direction. I wonder if we could ever say anything useful about cases where we stretch the {{MATH_27}}-axis by a different amount at each point? {{CMD_28}} Whao, that is a wild thought.