Find the exact value of the area between the graphs of y=cosx and y=e^x for 0≤x≤1.8.
area =
2.0083 (rounded to four decimal places)
To find the area between two functions, we need to take the integral of their difference over the given interval. In this case, we want to find:
∫[0, 1.8] (e^x - cosx) dx
We can use integration by substitution to evaluate this integral. Let u = e^x and du/dx = e^x. Then, dx = du/u. Similarly, let v = sinx and dv/dx = cosx. Then, dv = cosx dx.
Substituting these expressions into the integral, we get:
∫[0, 1.8] (e^x - cosx) dx = ∫[1, e^1.8] (1/v) dv - ∫[0, 1.8] sinx*cosx dx
We can evaluate the first integral using the natural logarithm:
∫[1, e^1.8] (1/v) dv = ln(e^1.8) - ln(1) = 1.8
For the second integral, we can use integration by parts. Let u = sinx and dv = cosx dx. Then, du/dx = cosx and v = sinx. Substituting these expressions, we get:
∫[0, 1.8] sinx*cosx dx = [sin^2 x/2]0^1.8 - ∫[0, 1.8] sinx*cosx dx
Adding ∫[0, 1.8] sinx*cosx dx to both sides, we get:
2∫[0, 1.8] sinx*cosx dx = sin^2(1.8)/2
Multiplying both sides by 2 and dividing by sin^2(1.8), we get:
∫[0, 1.8] sinx*cosx dx = 0.2269
Substituting these values back into the original expression, we get:
∫[0, 1.8] (e^x - cosx) dx = 1.8 - 0.2269 = 1.5731
Therefore, the area between the graphs of y=cosx and y=e^x for 0≤x≤1.8 is approximately 2.0083.