Calculus

Arc Length and Surface Area

Arc Length and Surface Area - Problem Solving

         

What is the area of the surface obtained by rotating the circle x2+y2=152x^2+y^2=15 ^2 about the line y=15?y=15 ?

Hint: You can use dua2u2=sin1ua+C, a>0.\int \frac{du}{\sqrt{a^2-u^2}}=\sin^{-1} \frac{u}{a}+C,\ a > 0.

What is the length of the curve represented by the equation x2/3+y2/3=1?\displaystyle x^{2/3}+y^{2/3}=1?

What is the length of the curve y=3xt31dt\displaystyle y=\int_{3}^{x} \sqrt{t^3-1} \,dt in the domain 1x4?1 \leq x \leq 4 ?

What is the area of the surface obtained by rotating the loop of the curve 48y2=x(16x)248y^2=x(16-x)^2 about the xx-axis?

A delivery drone flying at constant speed 15 m/s15 \text{ m/s} and constant height 2700 m2700 \text{ m} toward a destination drops its goods. If the trajectory of the falling goods until it hits the ground can be described by the equation y=2700x275,y=2700-\frac{x^2}{75}, where xx is the horizontal distance it travels and yy is its height above the ground, what is the distance (not horizontal displacement) traveled by the goods until it hits the ground?

Note: You can use a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C.\displaystyle \int \sqrt{a^2+u^2}\,du=\frac{u}{2}\sqrt{a^2+u^2}+\frac{a^2}{2}\ln(u+\sqrt{a^2+u^2})+C.

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