Volume by Cross Sections Section 7.2A Calculus AP/Dual, Revised ©2017 viet.dang@humble.k12.tx.us 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Review of Area How does one find area of a square? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Review of Area How does one find area of a semi-circle? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Review of Area How does one find area of a semi-circle? (or think of it this way) 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Review of Area How does one find area of an isosceles triangle? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Review of Area How does one find area of an equilateral triangle? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Calculus in Motion http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htm http://www.youtube.com/watch?v=3yIW9FqKXKU 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Slicing it Up 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Slicing it Up 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Equation 𝑨 𝒙 is the area of the cross sections (2 dimensions) 𝒅𝒙 is the with in respects to the axis 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Cross Sections Method For cross sections of area 𝑨 𝒙 taken perpendicular to the 𝒙-axis: 𝑽= 𝒂 𝒃 𝑨 𝒙 𝒅𝒙 For cross sections of area 𝑨 𝒚 taken perpendicular to the 𝒚-axis: 𝑽= 𝒄 𝒅 𝑨 𝒚 𝒅𝒚 Area Formulas: Square: 𝑨= 𝒔 𝟐 Semicircle: 𝑨= 𝝅 𝒅 𝟐 𝟖 Equilateral Right Triangle: 𝑨= 𝒔 𝟐 𝟑 𝟒 Isosceles Right Triangle: 𝑨= 𝒔 𝟐 𝟐 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections 4 Corner Example Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are ______________. What is the volume of the solid? Squares | Semi Circles | ----------------------------------------------------------------------------------------------- Isosceles Right Triangle | Equilateral Right Triangle 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Intersection Point Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are ______________. What is the volume of the solid? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections 4 Corner Model – Squares Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are SQUARES. What is the volume of the solid? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

4 Corner Model – Semi Circles Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are SEMICIRCLES. What is the volume of the solid? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

4 Corner Model – Isosceles Right Triangles Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are Isosceles Right Triangles. What is the volume? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

4 Corner Model – Equilateral Right Triangles Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are Equilateral Right Triangles. What is the volume? 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Steps Find the intersection points Determine the cross section and what axis is it perpendicular to Apply the PROPER cross section Evaluate 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 1 (with Calc) Find the volume of the solid whose base is the region bounded by the graphs of 𝒇(𝒙)= 𝒙  and 𝒈(𝒙)= 𝒙 𝟐 𝟖 , if slices perpendicular to the base along the x-axis have cross sections that are squares. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 1 (with Calc) Find the volume of the solid whose base is the region bounded by the graphs of 𝒇(𝒙)= 𝒙  and 𝒈(𝒙)= 𝒙 𝟐 𝟖 , if slices perpendicular to the base along the x-axis have cross sections that are squares. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 2 (Non-Calc) The base is the triangle enclosed by 𝒙+𝒚=𝟒, the 𝒙-axis, and 𝒚-axis. The slices are made perpendicular to the 𝒚-axis are semicircles. Find the volume of the figure. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 2 (Non-Calc) The base is the triangle enclosed by 𝒙+𝒚=𝟒, the 𝒙-axis, and 𝒚-axis. The slices are made perpendicular to the 𝒚-axis are semicircles. Find the volume of the figure. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 3 Find the volume of a solid whose base is bounded by the graphs 𝒚=𝒙 𝟑 , 𝒚=𝟎, and 𝒙=𝟏. Cross sections are perpendicular to the 𝒚-axis and are equilateral triangles. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Your Turn Find the volume of the solid whose base is the graphs, 𝒚=𝒙 and 𝒚 𝟐 =𝒙 and the cross sections taken are perpendicular to the 𝒙−axis are semicircles. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Steps Find the intersection points Determine the cross section and what axis is it perpendicular to Apply the PROPER cross section Evaluate 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 4 Find the volume of the solid whose base is the region inside the circle, 𝒙 𝟐 + 𝒚 𝟐 =𝟒 and the cross sections taken are perpendicular to the 𝒙−axis are squares. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 4 Find the volume of the solid whose base is the region inside the circle, 𝒙 𝟐 + 𝒚 𝟐 =𝟒 and the cross sections taken are perpendicular to the 𝒙−axis are squares. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 5 The base of a solid is the region enclosed by a circle centered at the origin and the radius is 5 inches. Setup the equation to find the volume of the solid if all of the cross sections are perpendicular to the 𝒙-axis are squares. Then, use a calculator to solve for the volume. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 5 (calc) The base of a solid is the region enclosed by a circle centered at the origin and the radius is 5 inches. Setup the equation to find the volume of the solid if all of the cross sections are perpendicular to the 𝒙-axis are squares. Then, use a calculator to solve for the volume. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Your Turn A solid has as its base the region in the 𝒙𝒚-plane bounded by the graph of 𝒙 𝟐 + 𝒚 𝟐 =𝟗. Find the volume of the solid if every cross section by a plane perpendicular to the 𝒙-axis is an equilateral triangle with base in the 𝒙𝒚-plane. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 6 (Non-Calc) Let 𝑹 be the region in the first quadrant bounded by the graph of 𝒚=𝟐 𝒙 , the horizontal line 𝒚=𝟔 and the 𝒚−axis. For each 𝒚, where 𝟎≤𝒚≤𝟔, the e cross section of the solid taken perpendicular to the 𝒚-axis is a rectangle whose height is 3 times the length of its base in region 𝑹. Write, but do not evaluate, an integral expression that gives the volume of the solid. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 6 (Non-Calc) Let 𝑹 be the region in the first quadrant bounded by the graph of 𝒚=𝟐 𝒙 , the horizontal line 𝒚=𝟔 and the 𝒚−axis. For each 𝒚, where 𝟎≤𝒚≤𝟔, the cross section of the solid taken perpendicular to the 𝒚-axis is a rectangle whose height is 3 times the length of its base in region 𝑹. Write, but do not evaluate, an integral expression that gives the volume of the solid. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 7 (calc) Find the volume of the solid whose base is the region is bounded by, 𝒙 𝟐 =𝟏𝟔𝒚 and 𝒚=𝟐. Cross section by a plane perpendicular to the 𝒙-axis are rectangles whose height are TWICE the side of the base in the 𝒙𝒚-plane. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 7 (calc) Find the volume of the solid whose base is the region is bounded by, 𝒙 𝟐 =𝟏𝟔𝒚 and 𝒚=𝟐. Cross section by a plane perpendicular to the 𝒙-axis are rectangles whose height are TWICE the side of the base in the 𝒙𝒚-plane. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 7 (calc) Find the volume of the solid whose base is the region is bounded by, 𝒙 𝟐 =𝟏𝟔𝒚 and 𝒚=𝟐. Cross section by a plane perpendicular to the 𝒙-axis are rectangles whose height are TWICE the side of the base in the 𝒙𝒚-plane. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 8 (calc) Let 𝑹 be the region bounded by the graph, 𝒚= 𝐥𝐧 𝒙 𝟐 +𝟏 , the horizontal line of 𝒚=𝟑 and vertical line of 𝒙=𝟏. Cross sections are perpendicular to the 𝒙-axis is a triangle that is twice the length of the base. Find the volume. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Example 8 (calc) Let 𝑹 be the region bounded by the graph, 𝒚= 𝐥𝐧 𝒙 𝟐 +𝟏 , the horizontal line of 𝒚=𝟑 and vertical line of 𝒙=𝟏. Cross sections are perpendicular to the 𝒙-axis is a triangle that is twice the length of the base. Find the volume. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Your Turn Find the volume of the solid that lies between planes of 𝒚= 𝒙 , 𝒚= 𝒆 −𝒙 and the 𝒚−𝒂𝒙𝒊𝒔. The cross sections perpendicular to the 𝒙-axis is a semicircle. 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections

7.2A: Volume using Cross Sections Assignment Worksheet 11/18/2018 3:04 AM 7.2A: Volume using Cross Sections