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5. (12 points) CAPE, which stands for Convectionle Potential Energy, is the area between a temperature sounding and the local moist adiabat that a lifted air parcel follows between the Level of Free Convection (LFC) and the Equilibrium Level (EL). In z (height) coordinates, CAPE can be found by the definite integral .EL CAPE = 9. (Ip (2 ) - Te(z) dz Te (z) (1) where To(z) is the air parcel temperature (in Kelvin K ) at height z m, Te(z) is the environmental temperature (in K ) at height z m, g is the gravitational constant (9.8 m/s?), and LFC and EL are the heights (in meters) of the level of free convection and equilibrium level respectively. August 10-11, 2020 saw a powerful derecho affecting eastern Nebraska, Iowa, Illinois, Wisconsin, and Indiana. It caused high winds and spawned an outbreak of weak tornadoes. Below is data for Omaha from August 10, 2020. Here, LFC = 3600 m and EL = 13, 200 m. In the table, we have defined f(z) = g. (p(z) - Te(z) Te (2) and computed these values in the last column. Elevation z (m) Tp (z ) ( K) Te (2) (K) f(z) 3600 m 283 283 0 5200 m 275 268 0.256 6800 m 266 258 0.304 8400 m 256 246 0.319 10,000 m 246 235 0.459 11,600 m 227 224 0.131 13,200 m 216 216 0 CEL (a) Estimate the value of CAPE = / 9 . (Ip(z) - Te(2) ) dz by calculating the value of Te (2 ) T3. That is, use the trapezoid rule with n = 3. CEL (b) Estimate the value of CAPE = g . (Ip(z) - Te(z) ) dz by calculating the value of Te (z) M3. That is, apply the Midpoint Rule using 3 rectangles. (c) Use Simpson's Rule to estimate CAPE by calculating S6.First, let's use the power-reduction identity sin (x) = 1 - cos(2x) 2 to reduce the powers of sine. We can rewrite sin (x) as sin(x) . sin(x) and then apply the power-reduction identity to sin (x) which is equivalent to (sin 2 (x) ) 3 Next, we substitute u = sin(x) , SO du = cos(x)dx The integral then becomes an expression in terms of u , which is easier to integrate. Finally, we can use integration by parts (if necessary) to compute the integral. The formula for integration by parts is Judo = uv - fvdu Here's the detailed computation: 1. Rewrite the integral: [ sin' (x)cos3(x)dx = ] (sin(x) . sin(x) ) cos3(x)dx = ] (sin=(x) ) . sin(x) ) cos3(x)dx 2. Apply the power-reduction identity: 1 - cos(2x) 2 . sin(x) cos3 (x) dx 3. Substitute u = sin(x) du = cos(x)dx 2 - cos( 2u) . u cos? (u) du 4. Compute the integral using integration by parts if necessary