Applications of Integration

$\pi \int_{-1}^{1} (x^2)^2 dx$

$\pi \int_{0}^{1} (x^4) dx$

$\int_{-1}^{1} x^2 dx$

$\int_{-1}^{1} \pi x^4 dx$

The distance from the origin to the edge of the disc.

The height of the disc.

The slope of the tangent line to the function.

The circumference of the disc.

Cube

Cylinder

Cone

Sphere

The radius of curvature for each micro-section

The linear density distribution across each cross-section

A differential element along axis rotation ($dx$ or $dy$)

A constant value representing slice separations

By multiplying $\pi$ by the square of the radius.

By taking the derivative of the function.

By integrating the function over a specific interval.

By dividing the region into rectangular sections.

Decrease value 'a'

Increase value 'b'

Increase or decrease depending on position along curve relation '+'

Maintain current value 's'

Shell method with respect to x.

Cross-sections perpendicular to the x-axis.

Disc method with respect to y.

Washer method with respect to y.

$\pi \int_{-4}^{4} (16-y^2) , dy$

$\int_{-4}^{16} \pi (y^2+16)^2 , dy$

$\pi \int_{-4}^{4} (y^2) , dy$

$\pi \int_{-4}^{4} (16+y^2) , dy$

Hole diameter equals $\sqrt{2}$ times height

Distance across the narrowest portion measures equal to three times the thickness of the wall itself

Diameter lengthwise exceeds thrice the sum of the individual radii included in the segments

Total capacity equates to half the product of the squared base circumference and the height

$\frac{32\pi}{5}$

$\frac{64\pi}{3}$

$\frac{16\pi}{3}$

$16\pi$