Advanced Placement (AP), 16.12.2021 20:40 QueenB7591

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See diagram.we use washers to get the area and we will be integrating with respect to xouter radius is the distance between y = e^x and y = -1 $r_{\text{out} } = e^x ) = e^x + 1$ inner radius of washer is the distance between y = 1 and y = -1 $r_{\text{in} } = 1 - (-1) = 2$ the area of one cross-section of a washer is $\begin{aligned} a(x) & = \pi(r_{\text{out} } )^2 - \pi(r_{\text{in} } )^2 \\ & = \pi(e^x + 1 )^2 - \pi(2)^2 \\ & = \pi \big[(e^x + 1 )^2 - 4 \big] \end{aligned}$ solid lies between x = 0 and x = 2. the volume is $\begin{aligned} v & = \int_0^2 a(x) dx \\ & = \int_0^2 \pi \big[(e^x + 1 )^2 - 4 \big] dx \end{aligned}$ if you need this expanded, then $\begin{aligned} v & = \int_0^2 \pi \left(e^{2x} + 2e^x + 1 - 4 \right) dx \\ & = \int_0^2 \pi \left(e^{2x} + 2e^x -3 \right) dx \end{aligned}$ the question doesn't say to evaluate, so we're done.

Answer from: Quest

800 pounds

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Answer from: Quest

answer: following the publication of mysterium and with the blessing of the graz school inspectors, kepler began an ambitious program to extend and elaborate his work. he planned four additional books: one on the stationary aspects of the universe (the sun and the fixed stars); one on the planets and their motions; one on the physical nature of planets and the formation of geographical features (focused especially on earth); and one on the effects of the heavens on the earth, to include atmospheric optics, meteorology, and astrology.[24]

he also sought the opinions of many of the astronomers to whom he had sent mysterium, among them reimarus ursus (nicolaus reimers bär)—the imperial mathematician to rudolph ii and a bitter rival of tycho brahe. ursus did not reply directly, but republished kepler's flattering letter to pursue his priority dispute over (what is now called) the tychonic system with tycho. despite this black mark, tycho also began corresponding with kepler, starting with a harsh but legitimate critique of kepler's system; among a host of objections, tycho took issue with the use of inaccurate numerical data taken from copernicus. through their letters, tycho and kepler discussed a broad range of astronomical problems, dwelling on lunar phenomena and copernican theory (particularly its theological viability). but without the significantly more accurate data of tycho's observatory, kepler had no way to address many of these issues.[25]

instead, he turned his attention to chronology and "harmony," the numerological relationships among music, mathematics and the physical world, and their astrological consequences. by assuming the earth to possess a soul (a property he would later invoke to explain how the sun causes the motion of planets), he established a speculative system connecting astrological aspects and astronomical distances to weather and other earthly phenomena. by 1599, however, he again felt his work limited by the inaccuracy of available data—just as growing religious tension was also threatening his continued employment in graz. in december of that year, tycho invited kepler to visit him in prague; on january 1, 1600 (before he even received the invitation), kepler set off in the hopes that tycho's patronage could solve his philosophical problems as well as his social and financial one

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