Which of the following definite integrals gives the length of a smooth planar curve…

MATHEMATICS · Single correct · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

Which of the following definite integrals gives the length of a smooth planar curve described explicitly as

x=g(y)

for

c\le y\le d

? (a)

\displaystyle\int_{c}^{d} \sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}\,dy

(b)

\displaystyle\int_{c}^{d} \sqrt{1+\left(\dfrac{dx}{dy}\right)^{2}}\,dy

(c)

\displaystyle\int_{c}^{d} \sqrt{1+\left(\dfrac{dx}{dy}\right)^{2}}\,dx

(d)

\displaystyle\int_{c}^{d} \sqrt{\left(\dfrac{dx}{dy}\right)^{2}}\,dy