Find the values of c such that the area of the region bounded by the parabolas y = 4x2 − c2 and y = c2 − 4x2 is 4/3. (Enter your answers as a comma-separated list.)

Question

Find the values of c such that the area of the region bounded by the parabolas y = 4x² − c² and y = c² − 4x²

 is 4/3. (Enter your answers as a comma-separated list.)

Answer 1

Note that these functions are reflections and thus meet at y =0.

Thus, as c²-4x² faces downward, its integral on [-c/2,c/2] (or [c/2,c/2] where c is negative) is 1/2 * 4/3 =2/3

∫_-c/2^c/2 c²-4x² dx = c²x-4/3x³ |_-c/2^c/2 = c³/2-c³/6 - (-c³/2+c³/6) = 2/3c³ Then, if 2/3c³ = 2/3, c = 1

Thus, c = ±1