Final answer:
The area enclosed by the curves y=x²−100 and y=100−x² is found by integrating the difference of the functions after determining their points of intersection, which yields the area using definite integration.
Step-by-step explanation:
The area enclosed by the curves y=x² −100 and y=100−x² can be found using integration. To calculate the enclosed area, one must find the points of intersection of the two curves by setting them equal to each other and solving for x. After finding these points, integrate the upper curve minus the lower curve between these limits.
Setting x² −100 equal to 100−x², we get x² = 50, which gives us x = ±7.0711 (approximately). The enclosed area is then given by the integral from –√50 to √50 of (100−x²)−(x²−100) dx, which simplifies to the integral from –√50 to √50 of 200− 2x² dx. The result of this integral will give the area enclosed by the curves.