7.5 Area Between Curvesap Calculus
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For each problem, find the area of the region enclosed by the curves. You may use the provided graph to sketch the curves and shade the enclosed region. 5) y = −2x2 − 1 y = −x + 3 x = 0 x = 1 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 6) y = 2 3 x2 y = x x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 7) y. AP Calculus AB - Worksheet 57 Area Between Two Curves – y-axis Find the area of the shaded region analytically. 1) 2) 3) 4) 5) 6) 7) Find the area of the region(s.
Chapter 7.1 Area of Region Between Two Curves Always ∫ ∫or Steps to find Area between 2 Curves Step 1: Sketch curves Step 2: Draw/label representative rectangle (identify ) Step 3: Find intersection of curves (a and b) Step 4: Identify – simplify if you can. Be able to nd the area between the graphs of two functions over an interval of interest. Know how to nd the area enclosed by two graphs which intersect. PRACTICE PROBLEMS: 1. Let Rbe the shaded region shown below. (a) Set up but do not evaluate an integral (or integrals) in terms of xthat represent(s) the area of R. Figure 7.5 g, 0,x 1 f, NOTE The height of a representative rectangle is regardless of the relative position of the axis, as shown in Figure 7.4. X-f x g x Area of a Region Between Two Curves If and are continuous on and for all in then the area of the region bounded by the graphs of and and the vertical lines and is A b a f x g x dx.Show All NotesHide All NotesYou appear to be on a device with a ’narrow’ screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.Section 6-2 : Area Between Curves
*Determine the area below (fleft( x right) = 3 + 2x - {x^2}) and above the x-axis. Solution
*Determine the area to the left of (gleft( y right) = 3 - {y^2}) and to the right of (x = - 1). Solution
For problems 3 – 11 determine the area of the region bounded by the given set of curves.
*(y = {x^2} + 2), (y = sin left( x right)), (x = - 1) and (x = 2) Solution
*(displaystyle y = frac{8}{x}), (y = 2x) and (x = 4) Solution
*(x = 3 + {y^2}), (x = 2 - {y^2}), (y = 1) and (y = - 2) Solution
*(x = {y^2} - y - 6) and (x = 2y + 4) Solution
*(y = xsqrt {{x^2} + 1} ), (y = {{bf{e}}^{ - ,frac{1}{2}x}}), (x = - 3) and the y-axis. Solution
*(y = 4x + 3), (y = 6 - x - 2{x^2}), (x = - 4) and (x = 2) Solution
*(displaystyle y = frac{1}{{x + 2}}), (y = {left( {x + 2} right)^2}), (displaystyle x = - frac{3}{2}), (x = 1) Solution
*(x = {y^2} + 1), (x = 5), (y = - 3) and (y = 3) Solution
*(x = {{bf{e}}^{1 + 2y}}), (x = {{bf{e}}^{1 - y}}), (y = - 2) and (y = 1) SolutionArea between CurvesThe area between curves is given by the formulas below.Formula 1:
Area = (int_a^b {,left| {fleft( x right) - gleft( x right)} right|,dx} )for a region bounded above by y = f(x) and below by y = g(x), and on the left and right by x = a and x = b.Formula 2:
(int_c^d {,left| {fleft( y right) - gleft( y right)} right|,dy} )
for a region bounded on the left by x = f(y) and on the right by x = g(y), and above and below by y = c and y = d.Example 1:1
Find the area between y = x and y = x2 from x = 0 to x = 1.
(eqalign{{rm{Area}} &= int_0^1 {left| {x - {x^2}} right|dx} &= int_0^1 {left( {x - {x^2}} right)dx} &= left. {left( {frac{1}{2}{x^2} - frac{1}{3}{x^3}} right)} right|_0^1 &= left( {frac{1}{2} - frac{1}{3}} right) - left( {0 - 0} right) &= frac{1}{6}})Example 2:1
Find the area between x = y + 3 and x = y2 from y = –1 to y = 1.
(eqalign{{rm{Area}} &= int_{ - 1}^1 {left| {y + 3 - {y^2}} right|dy} &= int_{ - 1}^1 {left( {y + 3 - {y^2}} right)dy} &= left. {left( {frac{1}{2}{y^2} + 3y - frac{1}{3}{x^3}} right)} right|_{ - 1}^1 &= left( {frac{1}{2} + 3 - frac{1}{3}} right) - left( {frac{1}{2} - 3 + frac{1}{3}} right) &= frac{{16}}{3}})See also7.5 Area Between Curves Ap Calculus Formulas7.5 Area Between Curves Ap Calculus CalculatorArea under a curve, definite integral, absolute value rules
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For each problem, find the area of the region enclosed by the curves. You may use the provided graph to sketch the curves and shade the enclosed region. 5) y = −2x2 − 1 y = −x + 3 x = 0 x = 1 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 6) y = 2 3 x2 y = x x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 7) y. AP Calculus AB - Worksheet 57 Area Between Two Curves – y-axis Find the area of the shaded region analytically. 1) 2) 3) 4) 5) 6) 7) Find the area of the region(s.
Chapter 7.1 Area of Region Between Two Curves Always ∫ ∫or Steps to find Area between 2 Curves Step 1: Sketch curves Step 2: Draw/label representative rectangle (identify ) Step 3: Find intersection of curves (a and b) Step 4: Identify – simplify if you can. Be able to nd the area between the graphs of two functions over an interval of interest. Know how to nd the area enclosed by two graphs which intersect. PRACTICE PROBLEMS: 1. Let Rbe the shaded region shown below. (a) Set up but do not evaluate an integral (or integrals) in terms of xthat represent(s) the area of R. Figure 7.5 g, 0,x 1 f, NOTE The height of a representative rectangle is regardless of the relative position of the axis, as shown in Figure 7.4. X-f x g x Area of a Region Between Two Curves If and are continuous on and for all in then the area of the region bounded by the graphs of and and the vertical lines and is A b a f x g x dx.Show All NotesHide All NotesYou appear to be on a device with a ’narrow’ screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.Section 6-2 : Area Between Curves
*Determine the area below (fleft( x right) = 3 + 2x - {x^2}) and above the x-axis. Solution
*Determine the area to the left of (gleft( y right) = 3 - {y^2}) and to the right of (x = - 1). Solution
For problems 3 – 11 determine the area of the region bounded by the given set of curves.
*(y = {x^2} + 2), (y = sin left( x right)), (x = - 1) and (x = 2) Solution
*(displaystyle y = frac{8}{x}), (y = 2x) and (x = 4) Solution
*(x = 3 + {y^2}), (x = 2 - {y^2}), (y = 1) and (y = - 2) Solution
*(x = {y^2} - y - 6) and (x = 2y + 4) Solution
*(y = xsqrt {{x^2} + 1} ), (y = {{bf{e}}^{ - ,frac{1}{2}x}}), (x = - 3) and the y-axis. Solution
*(y = 4x + 3), (y = 6 - x - 2{x^2}), (x = - 4) and (x = 2) Solution
*(displaystyle y = frac{1}{{x + 2}}), (y = {left( {x + 2} right)^2}), (displaystyle x = - frac{3}{2}), (x = 1) Solution
*(x = {y^2} + 1), (x = 5), (y = - 3) and (y = 3) Solution
*(x = {{bf{e}}^{1 + 2y}}), (x = {{bf{e}}^{1 - y}}), (y = - 2) and (y = 1) SolutionArea between CurvesThe area between curves is given by the formulas below.Formula 1:
Area = (int_a^b {,left| {fleft( x right) - gleft( x right)} right|,dx} )for a region bounded above by y = f(x) and below by y = g(x), and on the left and right by x = a and x = b.Formula 2:
(int_c^d {,left| {fleft( y right) - gleft( y right)} right|,dy} )
for a region bounded on the left by x = f(y) and on the right by x = g(y), and above and below by y = c and y = d.Example 1:1
Find the area between y = x and y = x2 from x = 0 to x = 1.
(eqalign{{rm{Area}} &= int_0^1 {left| {x - {x^2}} right|dx} &= int_0^1 {left( {x - {x^2}} right)dx} &= left. {left( {frac{1}{2}{x^2} - frac{1}{3}{x^3}} right)} right|_0^1 &= left( {frac{1}{2} - frac{1}{3}} right) - left( {0 - 0} right) &= frac{1}{6}})Example 2:1
Find the area between x = y + 3 and x = y2 from y = –1 to y = 1.
(eqalign{{rm{Area}} &= int_{ - 1}^1 {left| {y + 3 - {y^2}} right|dy} &= int_{ - 1}^1 {left( {y + 3 - {y^2}} right)dy} &= left. {left( {frac{1}{2}{y^2} + 3y - frac{1}{3}{x^3}} right)} right|_{ - 1}^1 &= left( {frac{1}{2} + 3 - frac{1}{3}} right) - left( {frac{1}{2} - 3 + frac{1}{3}} right) &= frac{{16}}{3}})See also7.5 Area Between Curves Ap Calculus Formulas7.5 Area Between Curves Ap Calculus CalculatorArea under a curve, definite integral, absolute value rules
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