Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.6.1

Chapter 6, Problem 6.6.1

What is the area of the curved surface of a right circular cone of radius 3 and height 4?

Verified step by step guidance

Step 1: Recall the formula for the curved surface area of a right circular cone, which is given by

A = πrl

, where

r

is the radius of the base and

l

is the slant height.

Step 2: To find the slant height

l

, use the Pythagorean theorem. The slant height is calculated as

l = √(r² + h²)

, where

r

is the radius and

h

is the height of the cone.

Step 3: Substitute the given values of

r = 3

and

h = 4

into the formula for the slant height:

l = √(3² + 4²)

. Simplify the expression inside the square root.

Step 4: Once the slant height

l

is determined, substitute

r = 3

and the calculated value of

l

into the curved surface area formula

A = πrl

Step 5: Simplify the expression for

A

to find the curved surface area of the cone. Leave the result in terms of

π

if required.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Surface Area of a Cone

The surface area of a right circular cone consists of two parts: the base area and the lateral (curved) surface area. The formula for the lateral surface area is given by πrl, where r is the radius and l is the slant height. The slant height can be calculated using the Pythagorean theorem, l = √(r² + h²), where h is the height of the cone.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is essential for finding the slant height of the cone in this problem.

Calculating Area

Calculating area involves determining the amount of space within a two-dimensional shape. For a cone, the area of the curved surface is calculated using the formula for lateral surface area. Understanding how to apply this formula correctly, along with the necessary dimensions, is crucial for solving the problem of finding the area of the cone's curved surface.

Recommended video:

05:23

Finding Area Between Curves on a Given Interval

Related Practice

Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = ln (x−√x²−1), for 1 ≤ x ≤ √2(Hint: Integrate with respect to y.)

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=x and y=4√x; about the x-axis

views

Textbook Question

Find the area of the region described in the following exercises.

The region bounded by y=|x−3|and y=x/2

Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = x³−x⁸+1,y=1; about the y-axis

views

Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

views

Textbook Question

Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)