Revised introduction to Numerical Integration methods.

tommytwoeyes · Feb 22, 2016 · d6bfacc · d6bfacc
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diff --git a/07_Integration_Techniques/.ipynb_checkpoints/Approximate_Integration-checkpoint.ipynb b/07_Integration_Techniques/.ipynb_checkpoints/Approximate_Integration-checkpoint.ipynb
@@ -6,7 +6,15 @@
    "source": [
     "# Approximate Integration\n",
     "\n",
-    "In some cases, it is impossible to find the exact value of a definite integral. In fact, the majority of functions in the real world have closed-form antiderivatives&mdash;that is, very few functions in the real world are composed of combinations of function families we have studied&mdash;algebraic functions, trigonometric functions, exponential functions, etc.\n",
+    "<p class='lead'>As beautiful &amp; efficient as the <em>Fundamental Theorem of Calculus</em> is, anyone interested in applying the tools of the calculus to real-world problems (scientists &amp; engineers, for example) will quickly find that additional tools are needed.</p>\n",
+    "\n",
+    "That is because, in some cases, it is impossible to find the exact value of a definite integral. In fact, _the majority of functions in the real world_ do not have closed-form antiderivatives&mdash;that is, very few functions in the real world are composed of _elemental functions_&mdash;combinations of function families we have studied, such as algebraic functions, trigonometric functions, and exponential functions.\n",
+    "\n",
+    "Fortunately, there is a viable alternative to using the Fundamental Theorem of Calculus to find definite integrals. \n",
+    "\n",
+    "You can _approximate_ definite integrals using simple, arithmetic methods similar to __Riemann Sums.__\n",
+    "\n",
+    "Essentially, these methods provide approximate values of the area under a curve between two endpoints (the limits of integration).\n",
     "\n",
     "#### Examples\n",
     "These functions cannot be integrated using the techniques of integration we've learned:\n",

diff --git a/07_Integration_Techniques/Approximate_Integration.ipynb b/07_Integration_Techniques/Approximate_Integration.ipynb
@@ -6,7 +6,15 @@
    "source": [
     "# Approximate Integration\n",
     "\n",
-    "In some cases, it is impossible to find the exact value of a definite integral. In fact, the majority of functions in the real world have closed-form antiderivatives&mdash;that is, very few functions in the real world are composed of combinations of function families we have studied&mdash;algebraic functions, trigonometric functions, exponential functions, etc.\n",
+    "<p class='lead'>As beautiful &amp; efficient as the <em>Fundamental Theorem of Calculus</em> is, anyone interested in applying the tools of the calculus to real-world problems (scientists &amp; engineers, for example) will quickly find that additional tools are needed.</p>\n",
+    "\n",
+    "That is because, in some cases, it is impossible to find the exact value of a definite integral. In fact, _the majority of functions in the real world_ do not have closed-form antiderivatives&mdash;that is, very few functions in the real world are composed of _elemental functions_&mdash;combinations of function families we have studied, such as algebraic functions, trigonometric functions, and exponential functions.\n",
+    "\n",
+    "Fortunately, there is a viable alternative to using the Fundamental Theorem of Calculus to find definite integrals. \n",
+    "\n",
+    "You can _approximate_ definite integrals using simple, arithmetic methods similar to __Riemann Sums.__\n",
+    "\n",
+    "Essentially, these methods provide approximate values of the area under a curve between two endpoints (the limits of integration).\n",
     "\n",
     "#### Examples\n",
     "These functions cannot be integrated using the techniques of integration we've learned:\n",