silemus.
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  1. 4L'Hopital's rule, limit
  2. 6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2
  3. 16Implicit differentiation, folium
  4. 21Maximizing rectangle area
  5. 22Optimizing box volume
  6. 23Related rates, sphere volume
  7. 24Related rates, circle area
  8. 27Integral of ln⁡x\ln x by parts
  9. 28Integration by parts
  10. 30Integral of sin⁡2x\sin^2 x
  11. 35Area between x2x^2 and xx
  12. 37Volume by disk method
  13. 41Telescoping series sum
  14. 44Maclaurin series of cos⁡x\cos x
  15. 45Radius of convergence, ratio test
  16. 49Double integral, inner yy first
  17. 50Separable ODE dy/dx=2ydy/dx = 2y

49Double integral, inner yy first

Evaluate ∫01∫0xxy dy dx\displaystyle \int_0^1 \int_0^x xy\,dy\,dx.

Do the inner integral in y first, treating x as constant: ∫₀ˣ xy dy = x³/2. Then integrate x³/2 from 0 to 1 using the power rule to get x⁴/8, which evaluates to 1/8. Switching the order of integration confirms the same answer, 1/8.

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