Mathematics, 31.08.2019 07:10 amandasantiago2001
Find the area of the region bounded by the parabola y = 5x2, the tangent line to this parabola at (4, 80), and the x-axis.
Answers: 1
units.
,
![= \int\limits^2_0\5x^{2} } \, dx + \int\limits^4_2 {(5x^{2} -y)} \, dx \\= \int\limits^2_0 {5x^{2} } \, dx + \int\limits^4_2 {(5x^{2}-40x+160) } \, dx \\= [\frac{5x^{3}}{3} ]^2_0 + [\frac{5x^{3} }{3} - \frac{40x^{2} }{2} + 160x ]^4_2](/tpl/images/0214/3395/ace4e.png)
![= [\frac{5(2)^{3} }{2} - \frac{5(0)^{2} }{2} ] + [\frac{5(4)^{3} }{3} - \frac{40(4)^{2} }{2} + 160(4) - [\frac{5(2)^{3} }{3} - \frac{40(2)^{2} }{2} + 160(2) ]]](/tpl/images/0214/3395/a0837.png)
, the slope of the line tangent to the parabola will be 
. Because it passes through the point (4, 80), its equation is
. If you were to plot the two functions, you'll notice that for 
the area can be given by integrating the vertical distance between the parabola and the x-axis, while for 
the area is given by integrating the vertical distance between the parabola and the tangent line. To avoid having to compute two integrals, we'll integrate along the vertical axis.
, the tangent line lies "above" the parabola over the entire domain of integration, so the area would be given by the single integral
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Find the area of the region bounded by the parabola y = 5x2, the tangent line to this parabola at (4...
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