Tangent Line and Tangent Plane Approximations of Definite Integral

Oftentimes, it becomes necessary to find approximate values for definite integrals, since the majority cannot be solved through direct computation. The methods of tangent line and tangent plane approximation can be derived as methods of integral approximation in two and three-dimensional spaces, respectively. Formulas are derived for both methods, and these formulas are compared with existing methods in terms of efficiency and error.