Which of the following gives the area of the circle $x^{2}+y^{2}=1$ ?
A) $\int_{-1}^{1} \sqrt{1-x^{2}} d x$
B) $4 \int_{-1}^{1} \sqrt{1-x^{2}} d x$
C) $4 \int_{0}^{1} \sqrt{1-x^{2}} d x$
D) $2 \int_{0}^{1} \sqrt{1-x^{2}} d x$

Question

Quizplus · Accepted Answer

The equation $x^{2}+y^{2}=1$ represents a circle with radius 1 centered at the origin. The integral in option C, $4 \int_{0}^{1} \sqrt{1-x^{2}} dx$, calculates the area of one quarter of the circle (in the first quadrant) and then multiplies by 4 to get the area of the entire circle. This method correctly accounts for the symmetry of the circle and uses the integral to find the area under the curve $y = \sqrt{1-x^{2}}$, which represents the upper half of the circle.

Which of the Following Gives the Area of the Circle x2+y2=1x^{2}+y^{2}=1x2+y2=1

Which of the Following Gives the Area of the Circle $x^{2}+y^{2}=1$