To find the volume of a cave, we fit $\mathrm { X } , \mathrm { Y }$ and Z axes such that the base of the cave is in the XY-plane and the vertical direction is parallel to the Z-axis. The base is the region in the XY-plane bounded by the parabola $y ^ { 2 } = 1 - x$ and the Y-axis. Each cross-section of the cave perpendicular to the X-axis is a square. (a) Show how to write a definite integral that will calculate the volume of this cave. (b) Evaluate this definite integral. Is it possible to evaluate it without using a formula for indefinite integrals?
To find the volume of a cave, we fit $\mathrm { X } , \mathrm { Y }$ and Z axes such that the base of the cave is in the XY-plane and the vertical direction is parallel to the Z-axis. The base is the region in the XY-plane bounded by the parabola $y ^ { 2 } = 1 - x$ and the Y-axis. Each cross-section of the cave perpendicular to the X-axis is a square.\\
(a) Show how to write a definite integral that will calculate the volume of this cave.\\
(b) Evaluate this definite integral. Is it possible to evaluate it without using a formula for indefinite integrals?