Volume of a Right Circular Cylinder using Integration
In this section, we will discuss the volume of a cylinder using integration with solved examples. Let us start with the prerequired knowledge to understand the topic, the volume of a cylinder using integration. The volume of a threedimensional object is defined as the space occupied by the object in the 3dimensional space. The volume of a cylinder is the space enclosed by its surface and is equal to πR^{2}h if its radius R and height h are given.
1.  Volume of a Right Circular Cylinder using Integration Formula 
2.  How to Find the Volume of a Right Circular Cylinder 
3.  FAQs on the Volume of a Cylinder using Integration 
Volume of a Right Circular Cylinder using Integration Formula
For a right circular cylinder, if its height (h) and base radius (R) is given, then its volume using integration can be given as shown below.
Consider a volume element as shown in the figure at right, which is at distance z from the center of the cylinder and has thickness dz. The volume of this element is equal to its base area times the thickness.
Volume of element (dV) = Area of base of the element × Thickness = (πR^{2}) dz
Summing up the volume of all such elements will give us the volume of a cylinder. Therefore, the volume of a cylinder is equal to,
Volume of the cylinder = Sum of all volumes of all such elements = Definite Integral of the volume of this element from z=0 to z=h
\[ \text{Volume of a Right Circular Cylinder, V}
= \int_{z = 0}^{h} \pi R^2 \,dz = \pi R^2 \bigg[ z \bigg]_0^h = \pi R^2 \bigg[ h  0 \bigg] = \pi R^2h \]
Note: Volume of an oblique cylinder is same as that of volume of a right cylinder, but vertical height is taken instead of the cylinder's slant height.
How to Find the Volume of a Right Circular Cylinder?
As we learned in the previous section, the volume of a right circular cylinder is πR^{2}h. Thus, we follow the steps shown below to find the surface area of a sphere in terms of diameter.
 Step 1: Identify the radius of the cylinder and name it to be R; Identify the height of the cylinder and name it to be h.
 Step 2: Find the volume of a cylinder using the formula = πR^{2}h.
 Step 3: Represent the final answer in cube units.
Example
Find the volume a cylinder having radius= 7 units and height = 10 units. (Use π = 22/7)
Solution
Height of the cylinder (h) = 10 units
Radius of the cylinder (R) = 7 units
Volume of a cylinder = π R^{2} h = (22/7) (7)^{2} 10 = 22 × 7 × 10 = 1540 units^{3}
Answer: Volume of a cylinder = 1540 units^{3}.
Solved Examples

Example 1
Find the volume of a cylinder with radius = 3 units and height = 7 units. (Use π = 22/7)
Solution
Radius of the cylinder (R) = 3 units
Height of the cylinder (h) = 7 units
Volume of the cylinder = π R^{2} h = (22/7) × 3^{2} × 7 = 22 × 9 = 198 units^{3}Answer: The volume of the cylinder is 198 units^{3}.

Example 2
Find the volume of the oblique cylinder having an oblique angle of 60°, and slant height is 2 units, and radius is 2 units. (Use π = 3.14)
Solution
Slant Height of the Cylinder (L) = 2 units
Radius of the Cylinder (R) = 2 units
Oblique Angle of the Cylinder (θ) = 30°
Height of the Cylinder (h) = L cosθ = 2 cos60°= 2 × (1/2) = 1 unit
Volume of the oblique cylinder = π R^{2}h = (3.14) × 2^{2} × 1 = 12.56 units^{3}Answer: The volume of the oblique cylinder is 12.56 units^{3}.
FAQs on the Volume of a Cylinder using Integration
What Is a Cylinder?
A cylinder shape in mathematics is a threedimensional solid figure which consists of two circular bases connected with two parallel lines.
What Is a Hollow Cylinder?
A hollow cylinder is one that is empty from the inside and has some difference between the internal and external radius. In other words, it is a cylinder that is empty from the inside and has some thickness at the peripheral.
How Do You Find the Volume of a Hollow Cylinder if Only Base Area and Height Is Given?
The volume of a cylinder is the capacity of the cylinder which signifies the amount of any material it can hold or the amount of any material that can be immersed in it. From the definition, the volume of the cylinder = Base Area × Height
What Is the Total Surface Area of a Hollow Cylinder?
For a given hollow cylinder, with both outer radius and inner radius known and having height h,
Total surface area of a hollow cylinder = Lateral Surface Area + 2 × Base Area = 2 π (r + R) h + 2(π R^{2}  π r^{2}) = 2 π (r + R)(h + R  r) cubic units.
Where R is the outer radius, r is the inner radius and h is the height of the hollow cylinder.
What Is the Annular Ring of a Hollow Cylinder?
The 2D shape formed at the bottom of a hollow cylinder is called an annular ring, i.e. it is a region bounded by two concentric circles. The base area of the hollow cylinder is the area of the annular ring of the cylinder.
What Is an Oblique Cylinder?
The cylinder with parallel bases that are not aligned with each other is known as an oblique cylinder.
What Is the Difference Between the Right Circular Cylinder and the Oblique Cylinder?
A cylinder can be either an oblique or a right circular cylinder. In a right circular cylinder, the bases are parallel and congruent circles where each line segment of the lateral curved surface is perpendicular to the bases. An oblique cylinder is a cylinder that is not a right circular cylinder.